Paper Title: A non-autonomous p-Adic diffusion equation on time changing graphs.



Operator-based semantics for choice programs: is choosing losing? (full version)
    
Jesse Heyninck


Open Universiteit, the Netherlands

University of Cape Town, South-Africa

    August 5, 2024
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§ ABSTRACT 
Motivated by the recently proven presence of ultrametricity in physical models (certain spin glasses) and the very recent study of Turing patterns on locally ultrametric state spaces, first non-autonomous diffusion operators on such spaces, where finitely many compact p-adic spaces are joined by a graph structure, are studied, including their Dirichlet and van Neumann eigenvalues. Secondly, the Cauchy problem for the heat equation associated with these operators is solved, its solution approximated by Trotter products, and thirdly, the corresponding Feller property as well as the Markov property (a Hunt process) is established. 






§ INTRODUCTION


The Gibbs measure associated with certain mean-field spin glass models, including the Sherrington-Kirkpatrick model,  was conjectured to  have ultrametric support in the thermodynamic limit
<cit.>. This conjecture, known as the Parisi ultrametricity conjecture, was proved by Panchenko  under the condition that the so-called Ghirlanda-Guerra identities are satisfied <cit.>.
This conjecture (and now result) has motivated up to this day the study of random processes on ultrametric state spaces, the most studied type being  diffusion on the p-adic numbers, cf. e.g. <cit.> and many other research articles in the physical and mathematical literature.


Ultrametricity of a space means that it has a hierarchical structure, i.e. its points can be viewed as the boundary of a tree, and therefore its topology, as well as any physical process on it, are determined by this tree structure. The p-adic numbers form a convenient mathematical framework for studying ultrametric spaces by viewing those trees as part of the regular tree structure of the p-adic numbers—as long as the  vertex degrees do not exceed the bound given by the p-adic regularity.
However, if the state space is only locally ultrametric, then it seemed unclear how to mathematically model diffusion on such structures until Zúñiga-Galindo's idea of modelling interactions between those local ultrametric pieces appeared via an externally imposed graph structure on this ensemble of pieces which then can be viewed as disjoint compact open subsets of the p-adic number field ℚ_p <cit.>. He even argues that p-adic analysis is a natural tool for studying Turing patterns on mean-field models.
In his case, these local pieces were modelled as p-adic balls, and thus he was lead to define a p-adic Laplacian integral operator resembling the integral operator representation of the well-studied Vladimirov (or Vladimirov-Taibleson) operator, cf. <cit.>.


The study of non-autonomous diffusion on p-adic domains seems to have been given significantly less consideration in the literature.
This motivates the authors of this article to investigate diffusion processes on locally hierarchical structures whose interactions (given as weighted edges) are time-dependent. The vertices of the underlying time-dependent graph are fixed in this setting. After clarifying the construction of p-adic operators associated with finite graphs—the “dictionary” set up in <cit.> needs a correction—properties of a time-dependent version of the Zúñiga operators from <cit.> are studied, proving that the uniform continuity of the kernel function  implies the strong continuity of the new time-dependent operator (Theorem <ref>) which is necessary for proving the Feller property later. 
Dirichlet and van Neumann eigenvalues  are calculated (Theorem <ref>), whereby the corresponding boundary conditions are generalised from graph theory. Thereafter,
the non-autonomous Cauchy problem for  the heat equation associated with these operators is solved (Theorem <ref>), thereby approximating the solution using Trotter products, and then finding that the temporal discretisation error is in first order O(1/ne^1/n) (Corollary <ref>). In the end the Feller property (Theorem <ref>) and corresponding Markov property—a Hunt process—is established (Theorem <ref>).  






§ NON-AUTONOMOUS P-ADIC EQUATIONS FOR TIME-CHANGING GRAPHS





 §.§ Preliminaries


In order to fix notation, denote the field of p-adic numbers as ℚ_p. The usual p-adic absolute value on ℚ_p is written as ·_p. The additive group of ℚ_p being a locally compact abelian group means that it has a Haar measure, denoted as dx (if the current variable is x). It is invariant under translation by adding any fixed element of ℚ_p, used for integrating functions, and is normalised here by setting

    ∫_ℤ_pdx=1

where ℤ_p is the subring of ℚ_p consisting of elements with p-adic absolute value at most 1, i.e. ℤ_p is the p-adic unit ball.


    
    For each t∈ [0,∞), we take a matrix A(t)∈ℂ^n× n. If for every t∈ [0,∞) we assume that A(t) is a symmetric matrix, then we can view this matrix as an adjacency matrix of a weighted graph 𝒢_t with n nodes in a natural way. For every t, the vertex set of 𝒢_t can be embedded into ℚ_p. The fact that 𝒢_t has an unchanging set of nodes allows to fix an embedding for all t>0. In other words the only difference between 𝒢_t and 𝒢_s for s≠ t are the edge weights. The set of vertices being constant, allows to write: V(𝒢):=V(𝒢_t), for any t>0. 
Following <cit.>,    
    we fix the embedding 

    V(𝒢)→ℚ_p,

and fix N such that each vertex I∈ V(G) lies in a distinct p-adic ball of radius r=p^-N

    B_r(I)={x∈ℚ_p: |x-I|≤ r}

sufficiently small such that all these balls are disjoint in ℚ_p. Let 

    K_N =_I∈ V(𝒢)B_p^-N(I).

Each vertex of 𝒢 can be assumed to have the form 

    I=I_0+I_1 p+...+I_N-1p^N-1,

where each I_i∈{0,…,p-1} for i=0,1,2,… is a p-adic digit.




 §.§ Basis of L^2(K_N,ℂ)


We are interested in the Hilbert space L^2(K_N,ℂ) with inner product  

    ⟨ f, g ⟩ =∫_K_N f(x)  g(x)   dx

Let Ω(p^r|x-a|_p) denote the indicator function of the ball B_-r(a). If r=N and a=I∈ V(G), we write 

    Ω(p^N|x-I|_p)=:Ω_I,N(x)

The space of functions X_N=_ℂ{Ω_I,N(x)}≅ℂ^n is an n-domensional vector space and it will be denoted by X_N. 


The functions 

    Ψ_j,n,r(x)=p^r/2χ_p(p^-r-1jx)Ω(p^r|x-p^-rn|_p)

where r∈ℤ, j∈{1,...,p-1}, and n runs through a fixed set of representatives of ℚ_p / ℤ_p are called Kozyrev functions.  
ion. If Supp (Ψ_j,n,r)=B_-r(p^-rn) ⊂ B_-N(I), then |p^-rn|_p=|I|_p, with N≤ r. 

Proof. We have that p^-r≤ p^-N, and 

    |p^-rn-I|_p<p^-N |p^-N-rn-p^-NI|_p<1,

where p^-NI∈ℚ_p/ℤ_p, and therefore |p^-NI|_p>1. Hence, if p^-N-rn≠ p^-NI, we have

    1>|p^-N-rn-p^-NI|_p= max{|p^-N-rn|_p,|p^-NI|_p}>1.

This shows that we must have |p^-N-rn|_p=|p^-NI|_p, implying the assertion. 

Therefore an orthonormal basis of 

    ℒ_0^2(K_N)=f∈ L^2(K_N)|∫_K_Nf(x) dx=0

is the set of functions of the form 

    Ψ_j,I,r(x)=p^r/2χ_p(p^-r-1jx)Ω(p^r|x-I|_p),
 with r≥ N, j=0,...p-1 and I∈ V(𝒢). 




We have L^2(K_N,ℂ)=X_N ⊕ℒ_0^2(K_N).


Proof. 
This is shown in <cit.>, where orthogonality uses the well-known result:






    ⟨Ω_I,N,Ψ_j,J,r⟩=0,

for every r≥ N, j=0,...,p-1 and I,J∈ V(𝒢), cf. e.g. <cit.> or <cit.>. 

By the above proposition, the set of functions consisting of the functions p^-N/2Ω_I,N(x) and the Kozyrev functions of the form (<ref>) form an orthonormal basis of L^2(K_N,ℂ). 




 §.§ Two operators over L^2(K_N,ℂ)

For every matrix A∈ℂ^n× n we can define a bounded linear operator in the following way. We define 

    A(x,y)=p^N∑_I,J∈ V(𝒢)A_I,JΩ_I,N(x)Ω_J,N(y).

This function is a test function on K_N × K_N, and hence, A(x,y)∈ L^2(K_N× K_N, ℂ). This function gives rise to an integral operator in the usual way: 

    𝒜 u(x)=∫_K_N A(x,y)u(y)dy.

This operator is a bounded linear operator from L^2(K_N,ℂ) to itself. 

Let A∈ℂ^n× n and 𝒜 its integral operator of the form (<ref>). Then X_N^⊥⊂ Ker(𝒜), where X_N^⊥ is the orthogonal complement of X_N.
  
Proof. Note that 

    𝒜u(x)=p^N∑_I,J∈ V(𝒢) A_I,J⟨ u,Ω_J,N⟩Ω_I,N(x)∈ X_N.

If u∈ X_N^⊥, then ⟨ u, Ω_I,N⟩=0, therefore by the above equality 𝒜u=0.   

For every matrix A∈ℂ^n× n, and every u∈ℒ_0^2(K_N), 𝒜u=0. In particular the set Kozyrev functions of the form (<ref>) are contained in Ker (𝒜). 

Proof. By Proposition <ref> and equality (<ref>), the result follows. 

Recall that the Laplacian matrix L is defined as 

    L=A-(γ_Iδ_IJ)

where A is the adjacency matrix of a graph 𝒢, γ_I=∑_JA_I,J is the degree of the vertex I, and δ_I,J is the identity matrix.  
We can associate to L an operator of the form (<ref>) denoted by ℒ. By Corollary <ref>,
the operator ℒ only "acts" on the space X_N, that is the operator ℒ
defined
similarly as in (<ref>):

    ℒu(x)=∫_K_NL(x,y)u(y) dy

with

    L(x,y)=p^N∑_I,J∈ V(𝒢)(A_IJ-γ_Iδ_IJ)Ω_I,N(x)Ω_J,N(x)

is just a rewrite of the Laplacian matrix in a space of dimension n, namely X_N≅ℂ^n and does not give an outcome of functions in ℒ_0^2(K_N). In particular, the Kozyrev functions of the form (<ref>) are not associated to any non zero eigenvalue of the operator ℒ. Therefore the operator ℒ is not an adequate candidate for the p-adic Laplacian, because it gives the same information as the Laplacian matrix, but not more. 


This is important in order to define an analogous operator to the Laplacian operator in the p-adic case. In <cit.>  it is stated that the operator ℒ coincides with the operator defined by the integral

    𝕃u(x)=∫_K_NA(x,y) (u(y)-u(x))dy

in the space of test functions, but this is not true as the following computations shows: 
Note that

    𝕃u(x)
       =∫_K_NA(x,y) (u(y)-u(x))dy
    
       =p^N∑_I,JA_I,J⟨ u,Ω_J,N⟩Ω_I,N(x)-u(x)γ(x),

where

    γ(x)=∫_K_NA(x,y)dy=∑_I,JA_I,JΩ_I,N(x).

Then let f=Ω(p^r|x-a|_p) be the indicator function of a ball B_-r(a)⊂ B_-N(J_0), for some J_0∈ V(𝒢). Then 

    ℒf(x)   =p^N∑_I,JL_I,J⟨ f, Ω_J,N⟩Ω_I,N(x)
       =p^N∑_I,JL_I,Jp^-rδ_J_0,JΩ_I,N(x)
       =p^N-r∑_IL_I,J_0Ω_I,N(x)
       =p^N-r∑_IA_I,J_0Ω_I,N(x)-p^N-rγ_J_0Ω_J_0,N(x),

On the other hand, we have 

    𝕃f(x)   =∫_K_NA(x,y) (f(y)-f(x))dy
            
       =p^N∑_I,JA_I,J⟨ f(y),Ω_J,N(y)⟩Ω_I,N(x)-f(x)γ(x),
       =p^N∑_I,JA_I,Jp^-rδ_J_0,JΩ_I,N(x)-f(x)γ(x)
       =p^N-r∑_IA_I,J_0Ω_I,N(x)-Ω(p^r|x-a|_p)γ(x),

If B_-r(a)⊂ B_-N(J_0) is a proper inclusion, then we can take an x_0∈ B_-N(J_0) ∖ B_-r(a). Then

    ℒf(x_0)=-p^N-rγ_J_0,

and

    𝕃f(x_0)=0.

Here we supposed that A_I,I=0 for all I, i.e. that 𝒢 is a simple graph. It is clear that, if r=N and a=I, then equality holds, that is, the two operators coincide in the space X_N. Moreover, since  ℒ and 𝒜 (the operator associated to the adjacency matrix of 𝒢) are zero for all u∈ X_N^⊥, we have for such u that

    ℒu(x)=0,

and

    𝕃u(x)=𝒜u(x)-u(x)γ(x)=-u(x)γ(x).

There is another fundamental difference between these two operators, as the following results shows. 



The operator ℒ is compact.


Proof. It is clear that ℒ is a Hilbert-Schmidt operator, since it is zero for all the Kozyrev functions supported in K_N. It is known that all Hilbert-Schmidt operators are compact. 


The operator 𝕃:L^2(K_N,ℂ)→ L^2(K_N,ℂ) defined by

    𝕃u(x)=𝒜u(x)-u(x)γ(x),

where 

    γ(x)=∫_K_NA(x,y)dy=∑_I∈ V(𝒢)γ_I Ω_I,N(x),

is a bounded linear operator, and all the Kozyrev functions of the form (<ref>) are eigenfunctions.


Proof. 
In <cit.> it is shown that 𝕃 is compact. The eigenfunction property and corresponding eigenvalues of the Kozyrev wavelets are shown in <cit.>.
Notice that the claim in loc. cit. about
𝕃 being compact is false, cf. Corollary <ref> below.








𝕃 is not compact.



This is an immediate consequence of Proposition <ref>.


Notice that the non-compactness of 𝕃 was also observed in
<cit.>.



The operator 𝒜 is called the p-adic Zúñiga operator associated with the finite graph 𝒢, and
𝕃  is called the corresponding p-adic Zúñiga Laplacian. 





 §.§ Non-autonomous p-adic Zúñiga operators

For a time-changing graph 𝒢_t we define its p-adic Laplacian as the function t↦𝕃(t), where

    𝕃(t)=𝒜(t)-Γ(t)

where

    𝒜(t)u(x)=∫_K_NA(x,y,t)u(y)dy,

and 

    Γ(t)u(x)=u(x)γ(t)(x),

where the kernel function A(x,y,t) and the degree function γ(t)(x) are given by 

    A(x,y,t)=p^N∑_I,J∈ V(𝒢) A_IJ(t)Ω_I,N(x)Ω_J,N(y),

and

    γ(t)(x)=∑_I∈ V(𝒢)γ_I(t)Ω_I,N(x)

with 
γ_I(t)=∑_J∈ V(𝒢)A_IJ(t).


The operators 𝒜(t) and 𝕃(t) are called non-autonomous p-adic Zúñiga operators. Operator 𝕃(t) is also called non-autonomous p-adic Zúñiga Laplacian. They are associated with the time-changing graph 𝒢_t.



If t↦ A_I,J(t) is uniformly continuous for each I,J∈ V(𝒢) then the p-adic Laplacian t↦𝕃(t) of 𝒢_t  is strongly continuous, that is t↦𝕃(t)u is a continuous function from [0,∞) to L^2(K_N,ℂ) for every u∈ L^2(K_N,ℂ).

Proof. Let u∈ L^2(K_N,ℂ)  then

    ||𝒜(t)u-𝒜(s)u||_L^2^2=∫_K_N| ∫_K_N(A(x,y,t)-A(x,y,s))u(y)dy|^2dx.

For ε>0, there is a positive real number δ>0 such that 

    |t-s|<δ|A_IJ(t)-A_IJ(s)|<ε/n^2.

Hence

    |A(x,y,t)-A(x,y,s)|   ≤∑_IJ|A_IJ(t)-A_IJ(t)|
       ≤ε.

Therefore

    ||𝒜(t)u-𝒜(s)u||_L^2^2   ≤∫_K_N(∫_K_N|A(x,y,t)-A(x,y,s)||u(y)|dy)^2dx
       ≤ε^2 ∫_K_N(∫_K_N|u(y)|dy)^2dx
       ≤ε^2(K_N)||u||_L^2^2,

where the last inequality is due to Hölder's inequality. 
On the other hand, 

    |γ(t)(x)-γ(s)(x)|   =|∑_I∈
    V(ℐ)(γ_I(t)-γ_I(s))Ω_I,N(x)|
    
       ≤∑_I|∑_JA_IJ(t)-A_IJ(s)|
    
       ≤ε

Hence, 

    ||Γ(t)u-Γ(s) u||_L^2^2   ≤∫_K_N|γ(t)(x)-γ(t)(x)|^2|u(y)|^2dx
       ≤ε^2 ||u||_L^2^2

Finally, we have 

    ||𝕃(t)u-𝕃(s)u||_L^2   ≤ ||𝒜(t)u-𝒜(s)u||_L^2+||Γ(t)u-Γ(s) u||_L^2
       ≤ε||u||_L^2((K_N)^1/2+1)

 This ends the proof. In particular the proof of Theorem <ref> shows that t↦𝕃(t) is continuous in the uniform operator topology. 


For each t>0, 𝕃(t) is a generator of a contraction semigroup {e^s𝕃(t)}, associated to the Cauchy problem of the heat equation: 

    ∂ u/∂ s (s,x)=ℒ(t)u(s,x)
    
          u(0,x)=u_0(x)∈ L^2(K_n,ℂ)
         .

with t≥ 0. 
Now we present the initial value problem called the evolution system or the non autonomous abstract Cauchy problem

    ∂ u/∂ t (t,x)=𝕃(t)u(t,x)
    
          u(s,x)=u_0(x)∈ L^2(K_n,ℂ)
         .

In the next section we present some results of the theory of semigroups and non autonomous differential equations. 




 §.§ Dirichlet and von Neumann eigenvalues


Assume that the kernel function A(x,y,t) is symmetric in (x,y), and here also that it is an analytic function for t≥0. Define for fixed t≥0 the p-adic edge set of 𝒢(t) as:

    Ω_E(t)=(x,y)∈ K_N^2| A(x,y,t)≠ 0,

as well as the p-adic vertex boundary:

    δ_NS(t)=
    x∈ K_N∖ S|∃ y∈ S A(x,y,t)≠0

as well as the p-adic edge boundary of 𝒢(t):

    ∂_NS(t)=(x,y)∈Ω_E(t)| x∈ S∧ y∉ S

which could also be called the vertex and edge boundaries of the operator 𝒜(t) defined in Section <ref>. Further, define the compact open set

    Ω_S^*(t)=(Ω_E(t)∩ S× S)∪∂_NS(t)⊂ℚ_p^2

Now, let S⊂ K_N be a compact open subset. Then define:

    V(S):=I∈ V| B_p^-N(I)∩ S≠∅.

The space of functions satisfying the Dirichlet boundary condition for S is

    D_S^*(t)=
    f∈ L^2(K_N)|f(x)=0 for all x∈δ_NS(t),

and the space of functions satisfying the von Neumann boundary condition for S is

    N_S^*(t)=   
       
    f∈ L^2(K_N)|∀ x∈δ_NS(t)∫_(x,y)∈∂_NS(t)A(x,y,t)(f(x)-f(y)) dy=0
    ,

again with fixed t≥0.


In view of <cit.>, the von Neumann boundary condition can be formulated as

    D_∂ S(t),+f(x)=0

where 

    D_∂ S(t),+u(x)=
    ∫_(x,y)∈∂_NS(t)
    A(x,y,t)(f(x)-f(y)) dy

is an advection operator defined similarly as D_+^α from <cit.>, and imitating the normal derivative w.r.t. the boundary of S in the classical case of Riemann manifolds.


The first Dirichlet eigenvalue of 𝕃(t) (with respect to S) is defined as:

    λ_1,𝕃^(D)(t)
    =inf_f≠ 0 f∈ D_S^*(t)p^N∫_Ω_S^*(t)A(x,y,t)f(x)-f(y)^2 dx dy/∫_Sf(x)^2γ(x)(t) dx

The von Neumann eigenvalue of 𝕃(t) with respect to S is defined as:

    λ_S,𝕃^(N)(t)
    =inf_f≠ 0∫_Sf(x)γ(x) dx=0p^N∫_Ω_S^*(t)A(x,y,t)f(x)-f(y)^2 dx dy/∫_Sf(x)^2γ(x)(t) dx

where it is assumed that δ_NS(t)≠∅ for this instance of t≥0.


The Dirichlet and von Neumann eigenvalues of 𝒢(t) w.r.t. S will be denoted as

    λ_1,𝒢^(D)(t)  and  λ_S,𝒢^(N)(t),

respectively.



Let ψ S→ℂ be a Kozyrev wavelet. Then ψ∈ N_S^*(t), i.e. ψ satisfies the von Neumann boundary condition.



Assume that the Kozyrev wavelet ψ is supported in a ball B⊂ B_p^-N(I) for some vertex I∈ V(S). Then

    ∫_(x,y)∈∂_NS(t)   A(x,y,t)(ψ(x)-ψ(y)) dy
    
       =∫_(x,y)∈∂_NS(t)A(x,y,t) dy ψ(x)
    
       -∫_(x,y)∈∂_NS(t)A(x,y,t)ψ(y) dy
    
       =0-0=0

where the left term vanishes, because x∈δ_NS(t), and
the right integral vanishes, because ψ is a Kozyrev wavelet, and   from Kozyrev's theorem that

    ∫_K_Nψ(x) dx=0,

cf. e.g. <cit.> or <cit.>.



Let S⊆ K_N be compact open, and assume that
δ S(t) is non-empty for t∈ [0,a) with a>0. Then the following statements hold true:


  * λ_1,𝕃^(D)(t)≤minλ_1,𝒢^(D)(t),γ̂_S(t)≤1,

  * λ_S,𝕃^(N)(t)≤minλ_S,𝒢^(N)(t),γ̂_S(t),

where

    γ̂_S(t)=min_I∈ V(S)_∂ V(S)(t)(I)/∫_Sγ(t)(x) dx

and _∂ V(S)(t)(I)
is the degree of vertex I in the edge boundary ∂ V(S)(t), viewed as a subgraph of 𝒢(t),  for t∈[0,a).



The inequalities λ_1,𝕃^(D)(t)≤λ_1,𝒢^(D)(t) and λ_S,𝕃^(N)(t)≤λ_S,𝒢^(N)(t) are obvious. Now, let ψ be a Kozyrev wavelet supported in B_p^-r(I)⊂ B_p^-N(I)∩ S for some vertex I∈ V(S). 
It clearly satisfies the Dirichlet boundary condition,  and, by Lemma <ref>, also the von Neumann boundary condition.
Hence, the Dirichlet and the von Neumann eigenvalues are both less than or equal to:

    ∑_J∈δ V(S)(t)A_IJ/∫_Sγ(x)(t) dx
    =_∂ V(S)(t)(I)/∫_Sγ(x)(t) dx

where δ V(S)(t) is the vertex boundary of the set V(S)⊂ V in 𝒢(t). This now shows the first inequality in 1., and that of 2.
The remaining inequality in 1. is given in <cit.>. This proves the theorem.


The following examples show that the inequalities can be strict:


Let 𝒢(t) be the complete graph on three vertices a,b,c, and let S=B_p^-N(a)⊔ B_p^-N(b).
Then, according to <cit.>, λ_1,𝒢^(D)(t) is an eigenvalue of the submatrix of the graph-Laplacian of  𝒢(t) spanned by the vertices a,b.
This is the matrix

    L_S=[    1 -1/2; -1/2    1 ]

with eigenvalues 3/2 and 1/2.
Hence, λ_1,𝒢^(D)(t)=1/2. But since the degree of each vertex of 𝒢(t) equals 2, it follows that

    λ_1,𝕃^(D)(t)≤γ̂_S(t)=1/4<1/2=λ_1,𝒢^(D)(t)

for given t≥0.



Let 𝒢(t) be the complete graph on two vertices a,b, and let S=B_p^-N(a). Then any non-trivial function f constant on the vertex balls B_p^-N(a) and B_p^-N(b) satisfying the von Neumann boundary condition, must satisfy
f(a)= 0 and thus

    ∫_Sf(x)^2γ(x)(t) dx=0

This means that 

    λ_S,𝒢(t)^(N)(t)=∞

and consequently

    λ_S,𝕃^(N)(t)<λ_S,𝒢^(N)(t)

because γ̂_S(t)<∞.








§ NON AUTONOMOUS CAUCHY PROBLEM


Consider the p-adic non autonomous Cauchy problem for a finite graph.
Most of the underlying theory for this section is taken from
<cit.>.




 §.§ Review of the abstract Cauchy problem





Let A(t) be a family of operators with t∈ℝ on a Banach space X. The following problem is addressed:


    ∂ u/∂ t (t)=A(t)u(t)
    
          u(s)=x∈ X

with t,s ∈ℝ and t≥ s. The system (<ref>) is called (nACP).


A continuous function u:[s,∞)→ X is called a (strict) solution of (<ref>), if u∈ C^1([s,∞),X), u(t)∈ D(A(t)) for all t≥ s, u(s)=x, and ∂ u/∂ t=A(t)u for t≥ 0.

The solution of non-autonomous Cauchy problem on some Banach space X can be given by a so called evolution family which can be defined as follows, cf. <cit.>:

A family (U(t,s))_t≥ s of linear operators on a Banach space X is called an evolution family for (<ref>), if 

    
  * U(t,r)U(r,s)=U(t,s), U(t,t)=id, t≥ s ∈ℝ.
    
  * The mapping (s,t)↦ U(t,s) is strongly continuous. 
  * ||U(t,s)||≤ M e^ω(t-s) for some M≥ 1, ω∈ℝ, t≥ s ∈ℝ.
    
  * For s∈ℝ the regularity space
    
    Y_s:={y∈ X:[s,∞) ∋ t→ U(t,s)y  solves  (nACP)}


is dense in X, where U(t,s)y is the unique solution of (nACP). 

If there exists a solving evolution family, then (<ref>) is called well-posed. We have the following Theorem.




Let X a Banach space and for every t>0 let A(t) be a bounded linear operator on X. If the function t↦ A(t) is strongly continuous for 0≤ t<T
(that is, t↦ A(t)u is continuous for each u∈ X), then for every x∈ X the problem (<ref>) is well-posed.



<cit.>.


We can apply the above Theorem to the problem (<ref>) for time-changing graphs.

The system (<ref>) is well-posed.

Proof. This is a direct consequence of Theorem
<ref>. 


If we let U(t,s)x=u(t), where u(t) is the solution of (<ref>), then the following holds  true: 

    
  * ||U(t,s)||≤ e^∫_s^t ||𝕃(τ)||dτ.
    
  * (t,s)↦ U(t,s) is continuous in the uniform operator topology for s≤ t.
    
  * ∂ U(t,s)/∂ t= ℒ(t) U(t,s).
    
  * ∂ U(t,s)/∂ s=-U(t,s)ℒ(s). 

This is due to the fact that t↦𝕃(t) is continuous in the uniform norm topology, cf. <cit.>.

In general, the solution does not have a closed form, but in special cases it does, as the following example shows.

 Under the hypothesis of Theorem <ref>, assume that A(t) and ∫_s^t A(τ) dτ commute for each s,t. Then 

    U(t,s)=e^∫_s^tA(τ) dτ

with the integral ∫ A(τ) dτ being interpreted "elementwise". 



<cit.>.





 §.§ Reviewing  the autonomous case for graphs



First, we review how to solve the heat equation in the autonomous case:


    ∂ u/∂ t (x,t)=𝕃u(x,t)
    
          u(0,x)=u_0(x)∈ L^2(K_n,ℂ)
         ,

where 𝕃 is interpreted as 𝕃=𝕃(t_0) for some t_0≥0. The following Theorem allows us to apply the Fourier Method (or the separation of variables method) to solve the system (<ref>). 


[Zúñiga-Galindo, 2020]
The elements of the set: 

    (L) {-γ_I: I∈ V(𝒢)}

 are the eigenvalues of of 𝕃. The corresponding eigenfunctions are given by the following infinite set 

    {φ_μ/||φ_μ||_2: μ∈(L)}{Ψ_j,I,r:I∈ V(𝒢), j∈{1,...,p-1}, r∈ℤ, r≥ N},

where the functions φ_μ(x) are defined by: 

    φ_μ(x)=∑_J∈ V(𝒢)φ_μ(J)Ω(p^-N|x-J|_p)

and the vector (φ_μ(J))_J∈ V(𝒢) is an eigenvector of the (negative semi-definite) Laplacian matrix L associated with adjacency matrix A, corresponding to μ∈(L), and Ψ_r,I,j are the Kozyrev functions of the form (<ref>). Furthermore we have the following orthogonal decomposition 

    L^2(K_N,ℂ)=X_N
        
        ⊕ℒ_0^2(K_N).





<cit.>.


According to <cit.>, the Cauchy problem of the heat equation (<ref>), with t≥ 0 has a solution which is unique. The solution is given by the semigroup {e^t𝕃} generated by 𝕃. By the decomposition (<ref>), we propose a solution of the form 

    u(x,t)=∑_μ∈(L) c_μ(t)φ_μ(x)+∑_(Ψ_j,I,r) ⊂ K_N c_r,I,j(t)ψ_j,I,r(x),

where the infinite set {Ψ_j,I,r|(Ψ_j,I,r) ⊂ K_N} is described in Theorem <ref>. Hence, the solution is an infinite sum, and the convergence takes place in L^2(K_N). After inserting this into (<ref>) we obtain 

    ∂ c_μ/∂ t =μ c_μ
    ∂ c_r,I,j/∂ t=-γ_Ic_j,I,r .

where μ and γ_I are the eigenvalues associated to φ_μ and ψ_j,I,r, respectively.


This proves that 

    e^t𝕃u_0(x)=u(x,t)=∑_μ∈(L) C_μe^μ tφ_μ(x)+∑_(Ψ_j,I,r) ⊂ K_N C_j,I,re^-γ_I tψ_j,I,r(x).

with C_I and C_j,I,r is uniquely determined by the initial conditions 

    u_0(x)=u(x,0)=∑_μ∈(L) C_μφ_μ(x)+∑_(Ψ_j,I,r) ⊂ K_N C_j,I,rψ_j,I,r(x),

and solution (<ref>) is an infinite convergent sum.





 §.§ Approximating the non-autonomous case for graphs


We now want to use the Trotter product formula to approximate the operator U(t,s), i.e., the solutions of (<ref>). As approximating operators we choose the product 

    ∏_k=1^n e^t/nℒ(s+kt/n)=e^t/nℒ(s+t)· e^t/nℒ(s+t-t/n)… e^t/nℒ(s+t/n).

We have the following proposition:


Let {A(t)}_t∈ℝ be a family of bounded and strongly continuous operators on a Banach space X, and let U(t,s) be its respective evolution family. Then 

    lim_n→∞∏_k=1^ne^t/nA(s+kt/n)x=U(s+t,s)x

for all x∈ X and uniformly for s and t in compact intervals of ℝ and ℝ_+, respectively. 



<cit.>.



Now we outline how to apply Proposition <ref> to approximate the solution of (<ref>). For 0<b, and ε>0, we have 

    ||e^t/n𝕃(t)· e^t/n𝕃(t-t/n)… e^t/n𝕃(t/n)u_0(x)-U(t,0)u_0(x)||_L^2<ε

with 0 ≤ t<b, and some n∈ℕ. Take 0≤ t_0 <b fixed, hence, in order to compute the approximate (respect to the L^2 norm) solution we have to solve (<ref>) by iterating the initial conditions, or in other words we have to compute e^t_0/n𝕃(t_0)· e^t_0/n𝕃(t_0-t_0/n)… e^t_0/n𝕃(t_0/n)u_0(x). We can view this approximation like snapshots of the system at n≥0 discrete time points. 
We call this the temporal discretisation, and we fix some notation:


    
  * Let μ(t_0) be an eigenvalue of the matrix L(t_0) and φ_μ(x,t_0) the associated eigenfunction.
    
  * Let {e^t𝕃(t_0)}_t≥ 0 be the semigroup generated by the operator 𝕃(t_0) for some t_0≥ 0. 
    
  * Let γ_I(t_0)=∑_JA_IJ(t_0)

Therefore, for a fixed t_0, the solution of the Cauchy problem in the autonomous case (<ref>) associated to the operator 𝕃(t_0), with initial condition u(x,0)=u_0(x) as in (<ref>), is given by 

    e^t𝕃(t_0)   u_0(x)=u(x,t)
    
       =∑_μ=μ(t_0)∈(L(t_0)) C_μ(t_0)e^μ(t_0) tφ_μ(x,t_0)
    
       +∑_r,I,j C_j,I,r(t_0)e^-γ_I(t_0) tψ_j,I,r(x).

Now it is clear that 

    e^t_0/n𝕃(t_0/n)u_0(x)
            =∑_μ=μ(t_0)∈(L(t_0)) C_μ(t_0/n)e^μ(t_0/n) t_0/nφ_μ(x,t_0/n)
         
            +∑_j,I,r C_r,I,j(t_0/n)e^-γ_I(t_0/n) t_0/nψ_j,I,r(x).

The next step is to compute e^t_0/n𝕃(2t_0/n)e^t_0/n𝕃(t_0/n)u_0(x). For this we need to find the solution to the Cauchy problem (<ref>) associated to the operator 𝕃(2t_0/n) with initial condition u_1(x)=e^t_0/n𝕃(t_0/n)u_0(x), and evaluate the solution t↦ e^t𝕃(2t_0/n)e^t_0/n𝕃(t_0/n)u_0(x) in t=t_0/n.


Define for a given adjacency matrix A(t) the corresponding weighted graph G_A(t) whose adjacency matrix is A(t). The corresponding L^2-space will  be denoted as L^2(G_A(t),ℂ). It is isomorphic to ℂ^V(𝒢).


To find the solution of the form (<ref>) we need to express the initial condition u_1(x)=e^t_0/n𝕃(t_0/n)u_0(x) as an element of L^2(G_A(2t_0/n),ℂ)⊕ℒ_0^2(K_N), 
that is 

    e^t_0/n𝕃(t_0/n)u_0(x)=∑_μ∈(L(t_0)) B_μφ_μ(x,2t_0/n)+∑_j,I,r B_j,I,rψ_r,I,j(x).

Then, after computing the coefficients B_μ and B_j,I,r the desired solution will be given by 

    e^t𝕃(2t_0/n)   e^t_0/n𝕃(t_0/n)u_0(x)= u_1(x,t)
            
       =∑_μ∈(L(t_0)) B_μe^μ(2t_0/n) tφ_μ(x,2t_0/n)+∑_j,I,r B_j,I,re^-γ_I(2t_0/n) tψ_j,I,r(x).

Comparing (<ref>) and (<ref>), it is clear that B_j,I,r=C_j,I,re^-γ_I(t_0/n)t_0/n. This is due to the fact that in decomposition (<ref>) the space ℒ_0^2(K_N) remains constant in time. Now we need to compute the constants B_μ. For this, note that 

    ω:=∑_μ∈(L(t_0)) B_μφ_μ(x,2t_0/n)=∑_μ∈(L(t_0)) C_μe^μ(t_0/n) t_0/nφ_μ(x,t_0/n)

Note that  for each s≥ 0, the set 

    φ_μ(x,s)=φ_μ(s)(x)|μ(s)∈(L(s))

is a basis for the finite dimensional space X_N. Hence, the coefficients B_μ are given by the basis change matrix from the basis {φ_μ(x,t_0/n)} to the basis {φ_μ(x,2t_0/n)}. The basis change matrix M=[M_λ,μ] is given by the following equalities: 

    φ_μ(x,2t_0/n)=∑_λ∈ V(𝒢)M_λ,μφ_λ(x,t_0/n).

In general, the basis change matrix from the basis {φ_μ(x,s)} to the basis {φ_μ(x,t)} M will be given by 

    M=M(s)^-1M(t),

where the matrices M(τ) are the modal matrices of the Laplacian matrix, that is, the μ-column of M(τ) is given by the eigenvector [φ_μ(K,τ)]_K∈ V(𝒢). Then, it is clear that the coefficients B_μ are given by

    [B_μ]_μ=M^-1[C_μe^μ(t_0/n) t_0/n]_μ.

Thus,

    [B_μ]_μ=M(2t_0/n)^-1M(t_0/n)(e^μ(t_0/n) t_0/n)[C_μ]_μ

and 

    B_j,I,r=C_j,I,re^-γ_I(t_0/n)t_0/n.

The process continues by solving the Cauchy problem for the operator 𝕃(3t_0/n) with initial condition e^t_0/n𝕃(2t_0/n)e^t/n𝕃(t_0/n)u_0(x). In order to do so, we repeat the process described above. That is we take as initial condition 

    u_1(x,t_0/n)   =∑_μ∈(L(t_0)) B_μe^μ(2t_0/n) t_0/nφ_μ(x,2t_0/n)
    
       +∑_j,I,r C_j,I,re^-γ_I(t_0/n)t_0/n-γ_I(2t_0/n) tψ_j,I,r(x)

and express it as an element of L^2(G_A(3t_0/n),ℂ)⊕ℒ_0^2(K_N).
Now,

    u_1(x,t_0/n)=∑_μ∈(L(t_0)) D_μφ_μ(x,3t_0/n)+∑_j,I,r D_j,I,rψ_r,I,j(x).

Again, we obtain the coefficients D_μ and D_j,I,r by:

    [D_μ]_μ=M(3t_0/n)^-1M(2t_0/n)(e^μ(2t_0/n)t_0/n)[B_μ]_μ

and 

    D_j,I,r=B_j,I,re^-γ_I(2t_0/n)t_0/n.

We can simplify these expressions by putting the value of B_μ and B_j,I,r in the above equalities, and we get 

    [D_μ]_μ=M(3t_0/n)^-1M(t_0/n)(e^μ(t_0/n)+μ_I(2t_0/n))[C_μ]_μ

and 

    D_j,I,r=C_j,I,re^-γ_I(t_0/n)t_0/n-γ_I(2t_0/n) t.

In the end, we obtain the function e^t_0/n𝕃(t_0)… e^t_0/n𝕃(t_0/n)u_0(x)∈ L^2(K_N), of the form 

    e^t_0/n𝕃(t_0)   · e^t_0/n𝕃(t_0-t_0/n)… e^t_0/n𝕃(t_0/n)u_0(x)=∑_μ∈(L(t_0))Q_μ(n,t_0)φ_μ(x,t_0)
       +∑_(Ψ_j,I,r)⊂ K_NC_r,I,j(t_0/n)e^-t_0/n∑_k=1^nγ_I(kt_0/n)Ψ_j,I,r(x),

where the coefficients Q_μ(n,t_0) are given by the equation

    [Q_μ(n,t_0)]_μ=M(t_0)^-1M(t_0/n)(e^t_0/n∑_k=1^nμ(kt_0/n))[C_μ(t_0/n)]_μ

For simplicity, we rename the approximation above: 

    U_n(t,0)u_0(x)   =∑_μ∈(L(t_0))Q_μ(n,t)φ_μ(x,t)
    
       +∑_(Ψ_j,I,r)⊂ K_NC_r,I,j(t/n)e^-t/n∑_k=1^nγ_I(kt/n)Ψ_j,I,r(x)

Then, for a general initial condition s, we obtain: 

    U_n(t,s)u_0(x)   =∑_μ∈(L(t_0))Q_μ(n,t,s)φ_μ(x,t_0)
    
       +∑_(Ψ_j,I,r)⊂ K_NC_r,I,j(s+t-s/n)e^-t-s/n∑_k=1^nγ_I(s+kt-s/n)Ψ_j,I,r(x),

where the coefficients Q_μ(n,t,s) are computed by the following equation: 

    [Q_μ(n,t,s)]=M(t)^-1M(s+t-s/n) (e^t-s/n∑_k=1^nμ(s+kt-s/n))[C_μ(s+t-s/n)]_μ

where the numbers C_μ(s+t-s/n) are the Fourier coefficients of the initial condition at time s+t-s/n. Note that in L^2(K_N) we have the following convergence:

    lim_n→∞U_n(t,s)u_0(x)=U(t,s)u_0(x),

and we can expand the function U(t,s)u_0(x)∈ L^2(K_N) by its Fourier series using the decomposition L^2(G_A(t_0),ℂ)⊕ℒ_0^2(K_N):

    U(t,s)u_0(x)=∑_μ∈(L(t_0))Q_μ(t,s)φ_μ(x,t_0)+∑_(Ψ_j,I,r)⊂ K_NQ_r,I,jΨ_j,I,r(x).

Hence, by Parseval's equality we obtain:

    lim_n→∞ Q_μ(n,t,s)=Q_μ(t,s)

and 

    lim_n→∞C_r,I,j(s+t-s/n)e^-t-s/n∑_k=1^nγ_I(s+kt-s/n)=C_r,I,j(s) e^-∫_s^tγ_I(τ)dτ,

where the last equality occurs by the continuity of t↦γ_I(t). On the other hand, the first limit  is more complicated, because it depends on the continuity of the matrix function t↦ M(t), more specifically, on the continuity of the eigenvectors: t↦φ_μ(t), where the vector φ_μ=[φ_μ(K,t)]_K is the eigenvector of the Laplacian matrix L(t) associated to μ(t)∈(L(t)). 

Let t↦ B(t) be a  matrix function. If B(t) is real symmetric, then we can find an orthonormal basis {φ_μ(t)}_μ  of ℂ^N formed by eigenvectors of B(t). It can be shown that if the matrix B(t) depends analytically on t inside a real interval, then we can chose an orthonormal basis {φ_μ(t)}_μ that depends analytically on t
<cit.>, 
and in particular, continuously on t. If  the analyticity assumption is deleted, the eigenvectors may not depend continuously on t. Therefore, if we suppose that the adjacency matrix A(t) depends analytically on t, then we have the following:


    lim_n→∞ [Q_μ(n,t,s)]_μ   =lim_n→∞M(t)^-1M(s+t-s/n) 
            
       ·(e^t-s/n∑_k=1^nμ(s+kt-s/n))[C_I(s+t-s/n)]_I
       =M(t)^-1M(s)(e^∫_s^tμ(τ)dτ)[C_μ(s)]_μ=[Q_μ(t,s)]_μ.

But, in order to have the above limit, it is sufficient to suppose that the eigenvectors φ_μ(t) are right continuous at the initial condition s≥ t_0. 


This proves the following:



If the eigenvectors of the Laplacian matrices L(t) are right continuous at some s≥0, then the evolution family of the Cauchy problem (<ref>) (with initial condition at s) is given by 

    U(t,s)u_0(x)   =∑_μ∈(L(t_0))Q_μ(t,s)φ_μ(x,t)
    
       +∑_(Ψ_j,I,r)⊂ K_N C_r,I,j(s) e^-∫_s^tγ_I(τ)dτΨ_j,I,r(x),

where t≥ s,

    [Q_μ(t,s)]=M(t)^-1M(s)(e^∫_s^tμ(τ)dτ)[C_μ(s)]_μ ,

and the coefficients C_μ(s) and C_r,I,j(s) are uniquely determined by the initial condition

    u_0(x)=∑_μ∈(L(t_0))C_μ(s)φ_μ(x,s)+∑_(Ψ_j,I,r)⊂ K_N C_r,I,j(s)  Ψ_j,I,r(x).

In particular, this result holds for all s≥ 0 if the adjacency matrix A(t) depends analytically on t.



Note that this is consistent with the autonomous problem, that is, when L(t)=L is constant. In this case, the matrices M(t) and the functions t↦μ(t), t↦φ_μ(t) are constant, thus the solution is given by 

    U(t,s)u_0(x)   =∑_μ∈(L)C_μ(s)e^(t-s)μ
       +∑_(Ψ_j,I,r)⊂ K_N C_r,I,j(s) e^-(t-s)γ_IΨ_j,I,r(x)

for t≥ s≥0.



Assuming analyticity of L(t), the first-order approximation error in the temporal discretisation is O(1/ne^1/n).



For the temporal discretisation, observe that by replacing each analytic function with its first order approximation leads to 
an error of O(1/n) in each entry of the eigenvectors and eigenvalues of L(t), and an error of O(e^1/n)
in each exponential integral, because the temporal discretisation is in this case a Riemann sum. Hence, the
error in each coefficient of the solution is given by the 
product error, and this is in first order O(1/n e^1/n), which proves the assertion. 



































§ INHOMOGENEOUS MARKOV PROCESSES AND FELLER EVOLUTION.





 §.§ Reviewing inhomogeneous Markov processes

In this section we review some definitions and results necessary to prove that the non-autonomous abstract Cauchy problem (<ref>) has attached to it an inhomogeneous Markov process. 


In the sequel E denotes a separable complete metrizable topological Hausdorff space. In other words E is a Polish space. The space C_b(E) is the space of all complex valued bounded continuous functions. The space C_b(E) is equipped with the uniform norm: ||·||_∞, and the strict topology 𝒯_β.

It is considered as a subspace of the bounded Borel measurable functions L^∞(E), also endowed with the supremum norm. 

Let (Ω,ℱ) be a measurable space. A filtration ℱ_t, t∈ [0,T], is a family of sub-σ-algebras of ℱ such that if 0≤ t_1 ≤ t_2 ≤ T, then ℱ_t_1⊂ℱ_t_2. A two-parameter filtration ℱ_t^τ, 0≤τ≤ t ≤ T, is a family of sub-σ-algebras of ℱ that is increasing in t and decreasing in τ. Let X_s, s∈ [0,T] be a stochastic process, with state space (E, ℰ). The σ-algebra σ(X_s:0≤ s≤ t) is the smallest σ-algebra such that all the state variables X_s with 0≤ s≤ t are measurable.

The process X_s, s∈ [0,T], generates the following two-parameter filtration: 

    ℱ_t^τ=σ(X_s:τ≤ s ≤ t)

with 0≤τ≤ t ≤ T. 


We recall some definitions:



A transition probability funcition P(r,x;s,A), where 0≤ r < s ≤ T, x∈ E, and A∈ℰ, is a nonnegative function for which the following condition hold: 

    
  * For fixed r,s and A, P is a nonnegative Borel measurable function on E. 
    
  * For fixed r,s and x, P is a Borel measure on ℰ. 
    
  * P(r,x;s,E)=1 for all r,s and x. 
    
  * P(r,x;s,A)=∫_EP(r,x;y,dy)P(u,y;s,A) for all r<u<s, and A. 


The number P(r,x;s,A) can be interpreted as the probability of the following event: The random system located at x∈ E at time r hits the target A⊂ E at time s. 



The process

    {
    (Ω,ℱ_T^τ,ℙ_τ,x), (X(t):T≥ t≥ 0),(E,ℰ)}

is called a time-inhomogeneous Markov process if 

    𝔼_τ,x[f(X(t))|ℱ_s^τ]=𝔼_s,X(s)[f(X(t))]

ℙ_τ,x-almost surely. Here f is a bounded Borel measurable function defined on the state space E and τ≤ s ≤ t ≤ T. 


Let P be a transition probability function. It is possible to construct a filtered measurable space (Ω, ℱ, ℱ_t^τ), a family of probability measures ℙ_τ,x, x∈ E, τ∈ [0,T], on the space (Ω, ℱ_T^τ), and a Markov process X_t, t∈ [0,T] such that 

    ℱ_t^τ=σ(X_s:τ≤ s ≤ t)

with 0≤τ≤ t ≤ T and 

    ℙ_τ,x(X_t∈ A)=P(τ,x;t,A)

with 0≤τ≤ t ≤ T, A∈ℰ.






A family {P(s,t):0≤ s ≤ t ≤ T} of operators defined on L^∞(E) is called a Feller Evolution or Feller propagator on C_b(E) if it possesses the following properties: 

    
  * It leaves C_b(E) invariant: P(s,t)C_b(E)⊂ C_b(E) for 0≤ s≤ t ≤ T;
    
  * It is an evolution: P(τ,t)=P(τ,s)P(s,τ) for all τ, s, t for which 0≤τ≤ s ≤ t and P(t,t)=I, t∈ [0,T];
    
  * If 0≤ f ≤ 1, f∈ C_b(E), then 0≤ P(s,t)f ≤ 1, for 0≤ s ≤ t≤ T. 
    
  * If the function (s,t,x)↦ P(s,t)f(x) is continuous on the space Λ:={(s,t,x)∈ [0,T]×[0,T]× E:s≤ t}. 


In the following theorem we see that with a Feller evolution a strong Markov process can be associated in such a way that the one-dimensional distributions or marginals are determined by the operators f↦ P(τ,t)f, f∈ C_b(E). In fact every operator P(τ,t) can be written as 

    P(τ,t)f(x)=∫ P(τ,x;t,dy)f(y)

for f∈ C_b(E). Where the mapping 

    (τ,x,t,B)↦ P(τ,x;t,B)

with (τ,x,t,B)∈ [0,T]× E× [0,T]×ℰ, τ≤ t, is a sub-probability transition function. 




Let {P(τ,t):τ≤ t ≤ T} be a Feller evolution in C_b(E). Then there exists a strong Markov process (in fact a Hunt Process) such that [P(τ,t)f](X)=𝔼_τ,x[f(X(t))], f∈ C_b(E), t≥0. Moreover, this Markov process possesses the properties in <cit.>. 



<cit.>.





 §.§ Stochastic process attached to a non-autonomous equation


In this section we prove that the system 

    ∂ u/∂ t (x,t)=𝕃(t)u(x,t)
    
          u(x,s)=u_s(x)∈ C(K_n,ℂ)

generates a time-inhomogeneous Markov process, where the operators 𝕃(t) satisfies the hypothesis of Theorem <ref>. In more generality, we study the following Cauchy problem 

    ∂ u/∂ t (x,t)=A(t)u(x,t)
    
          u(x,s)=u_s(x)∈ C(E) ,

where E is a compact Polish space. Thus, C_b(E)=C(E). Also, we suppose that t↦ A(t) is continuous in the uniform norm topology (In particular A(t) is bounded for all t≥ 0). Moreover, we suppose that for every τ≥ 0, A(τ) generates a strongly continuous, positive, contraction semigroup {e^tA(τ)}_t ≥ 0 on C(E). By Theorem <ref>,
the system (<ref>) 
is well-posed for every initial condition u_s(x)∈ C(E). Therefore, there exists an evolution family U(t,s), such that U(t,s)u_s(x) is the unique solution of the Cauchy problem (<ref>). 
Moreover, U(t,s) satisfies the properties in Remark <ref>.


The evolution U(t,s) described above is a Feller evolution.


Proof. 
We verify the properties of Definition <ref>.
The verification of 1. and 2. is straightforward.
We now prove condition 3. Let f∈ C(E) such that 0≤ f ≤ 1. By Proposition <ref>, 
we have 

    lim_n→∞∏_k=1^ne^(t-s)/n A(s+k(t-s)/n)f(x)=U(t,s)f(x)

in the supremum norm, and with 0≤ s ≤ t ≤ T. Therefore, we also have point-wise convergence. Since every semigroup {e^tA(τ)}_t≥ 0 is a strongly continuous, positive, contraction semigroup we have 

    0≤∏_k=1^n e^(t-s)/n A(s+k(t-s)/n)f(x)≤ 1.

We obtain the desire result by taking n→∞. For condition 4., note that, since the space E is Polish, we can prove the continuity in terms of sequences. Let (t,s,x)∈Λ, with s≤ t and x∈ E. And let (t_n,s_n,x_n) a convergent sequence, with s_n≤ t_n and x_n∈ E, such that (t_n,s_n,x_n)→(t,s,x) as n→∞. We have


    |U(t_n,s_n)f(x_n)-U(t,s)f(x)|   ≤ |U(t_n,s_n)f(x_n)-U(t,s)f(x_n)|
       +|U(t,s)f(x_n)-U(t,s)f(x) |  
    ≤   ||U(t_n,s_n)-U(t,s)||_∞||f||_∞
       +|U(t,s)f(x_n)-U(t,s)f(x) |

Since (t,s)↦ U(t,s) is continuous in the uniform norm topology (Remark 
<ref>.2.), 
and the function x↦ U(t,s)f(x) is continuous in E, by taking n→∞ the right hand side of the above inequality tends to zero. This now proves that U(t,s) is a Feller evolution. 
Let {A(t)}_t≥ 0 be a set of bounded operators on C(E). Suppose that t↦ A(t) is continuous in the uniform norm topology and that each A(t) generates a strongly continuous, positive, contraction semi-group. Then the Cauchy problem (<ref>)
is well-posed and its evolution family U(t,s) generates a strong Markov process with the properties listed in Theorem <ref>. 


Proof. This is a consequence of Lemma <ref>
and Theorem <ref>. 



The literature does not seem to contain a result proving that a non-autonomous Cauchy problem has attached to it a Markov process. That is why  the above result was obtained using the techniques learned above in order to find an aproximate solution (Proposition <ref>). 
In particular, it is important to note that evolution families are the two-parameter generalizations of semigroups, and analogously, Feller evolution families are the two parameter generalizations of Feller semigroups. 



There exists a probabilty transition function P(t,x;s,·), where (t,x,s)∈ [0,T]× K_N × [0,T], and s≤ t, on the Borel σ-algebra of K_N, such that the Cauchy problem (<ref>) 
has a unique solution of the form 

    u(x,t)=∫_K_N u_s(x) P(t,x;s,dy).

In addition, P(t,x;s,·) is the transition function of a Markov process with the properties listed in Theorem <ref>. 

Proof. We have to proove that t↦𝕃(t) satisfies the hypothesis in Theorem <ref>. 
By   <cit.>, we know that 𝕃(t) generates a strongly continuous, positive, contraction semigroup on C(K_N) for each t≥0. hence, we only have to show that t↦𝕃(t) is continuous in the uniform norm topology. Recall from the proof of Theorem
<ref> that 

    ||𝕃(t)u-𝕃(s)u||_∞≤ ||𝒜(t)u-𝒜(s)u||_∞+||Γ(t)u-Γ(s)u||_∞.

Let ε>0, and δ>0 such that 

    |t-s|<δ|A(x,y,t)-A(x,y,s)|<ε and   |γ(t)(x)-γ(s)(x)|< ε

We have that

    ||𝒜(t)u-𝒜(s)u||_∞   =sup_x∈ K_N|∫_K_N (A(x,y,t)-A(x,y,s))u(y)dy|
       ≤ε||u||_∞(K_N)

and 

    ||Γ(t)u-Γ(s)u||_∞ =sup_x∈ K_N|(γ(t)(x)-γ(s)(x))||u(x)|≤ε ||u||_∞ .

If ||u||_∞≤ 1, then

    ||𝕃(t)u-𝕃(s)u||_∞≤ε((K_N)+1).

This inequality implies the continuity in the uniform norm topology. We now apply Theorem <ref> and Theorem <ref> to obtain the result. 




§ ACKNOWLEDGEMENTS


The first named author is indebted to Norbert Paul for coming up  with questions on how to study topological questions in the setting of finite spaces a long time ago, which gave a long-lasting impulse to eventually combine these with ultrametric analysis and mathematical physics.
The second named author is indebted to
Wilson Zúñiga-Galindo for sharing valuable insights into p-adic analysis when he was under his supervision as an undergraduate student. Martin Breunig, Markus Jahn and David Weisbart are warmly thanked for fruitful discussions.

This work is supported by the Deutsche Forschungsgemeinschaft
under project number 469999674.



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 Paper Title: Asymptotic windings, surface helicity and their applications in plasma physics.









	Wadim Gerner[E-mail address: mailto:wadim.gerner@inria.frwadim.gerner@inria.fr]




Sorbonne Université, Inria, CNRS, Laboratoire Jacques-Louis Lions (LJLL), Paris, France


Abstract: 
In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper we investigate properties of surface helicity and in particular answer two open questions posed in the aforementioned work: (i) We give a precise mathematically rigorous physical interpretation of surface helicity in terms of linking of distinct field lines. (ii) We prove that surface helicity is non-trivial if and only if the underlying surface has non-trivial topology (i.e. at least one hole).

We then focus on toroidal surfaces which are of relevance in plasma physics and express surface helicity in terms of average poloidal and toroidal windings of the individual field lines of the underlying vector field which enables us to provide a connection between surface helicity and rotational transform. Further, we show how some of our results may be utilised in the context of coil designs for plasma fusion confinement devices in order to obtain coil configurations of particular "simple" shape.

Lastly, we consider the problem of optimising surface helicity among toroidal surfaces of fixed area and show that toroidal surfaces admitting a symmetry constitute global minimisers.
Keywords: Submanifold helicity, Surface helicity, Biot-Savart operator, Plasma physics, Coil winding surface, Rotational transform
2020 MSC: 14J81, 37E35, 49Q10, 78A30, 78A55


§ INTRODUCTION

	In the realm of 3-dimensional (magneto-)hydrodynamics there is a well-known preserved physical quantity called helicity. To set the stage, we let Ω⊂ℝ^3 be a bounded smooth domain and we let u be a smooth vector field which is divergence-free on Ω and tangent to its boundary. The 3-d Biot-Savart helicity of u may then be defined as follows
	
    ℋ(u):=1/4π∫_Ω×Ωu(x)·(u(y)×x-y/|x-y|^3)d^3yd^3x.

	An alternative way to express helicity consists of defining the Biot-Savart potential of u by 
    BS_Ω(u)(x):=1/4π∫_Ωu(y)×x-y/|x-y|^3d^3y

	so that ℋ(u) is just the L^2-inner product of u and BS_Ω(u), c.f. <cit.>. From a physical perspective helicity may be regarded as a measure of the average linking of distinct field lines of the underlying vector field u, see <cit.>,<cit.>,<cit.>. The relevance of helicity is that it is preserved under the action of volume preserving diffeomorphisms, c.f. <cit.>, <cit.>,<cit.> and in fact is in some sense the only such invariant <cit.>. In the context of the incompressible Euler equations the velocity field v satisfies the equations
	
    ∂_tv+∇_vv=∇ p, div(v)=0 in Ω, v∥∂Ω,

	where p denotes the pressure function. Upon applying the curl operator to the first equation one verifies that the vorticity ω of the fluid, i.e. the curl of the velocity field v, is a solution to
	
    ∂_tω=-[v,ω],
 
	where [·,·] denotes the Lie-bracket of vector fields. This implies that ω(t,x)=((ψ_t)_*ω_0)(x), where ψ_t denotes the flow of v, (ψ_t)_* denotes the pushforward with respect to ψ_t and ω_0(x):=ω(0,x)=curl(v(0,x)) is the initial vorticity. Since v∥∂Ω and div(v)=0 we conclude that ψ_t is well-defined and volume-preserving. We hence conclude from our preceding discussion that the helicity of the vorticity vector field is preserved in time. In the context of ideal magnetohydrodynamics, one can show in a similar spirit that the helicity of the underlying magnetic field is preserved in time. Even more, helicity also provides a lower bound on the L^2-norm of the underlying vector field, which in the context of ideal magnetohydrodynamics corresponds to the magnetic energy of the underlying magnetic field, c.f. <cit.>, <cit.>
	
    |ℋ(u)|≤ c(Ω)∫_Ω|u(x)|^2d^3x

where c(Ω)>0 is a constant independent of u.
It is further interesting to understand which other diffeomorphisms, beyond the volume-preserving diffeomorphisms, preserve helicity. This question has been investigated in <cit.> and it turns out that in general there exist non-volume preserving diffeomorphisms which preserve helicity.

In <cit.> Cantarella and Parsley introduced shortly the notion of submanifold helicity, a special case of which is the surface helicity, which corresponds to the (0,2,3)-helicity in the terminology of <cit.>. As we have seen, the notion of 3-d helicity is an important quantity in the context of fluid dynamics. It is therefore of interest to understand its 2-d analogue since vector fields tangent to surfaces appear naturally in many applications, some of which we will discuss in the next sections.
The main goal of the present manuscript is to illuminate our understanding of surface helicity and establish connections to other known dynamical quantities.
Outline of the paper: In the next section we present the main results which we divided in subsections to increase readability. We first discuss answers to some open questions raised in <cit.> regarding the surface helicity and provide in particular a general interpretation of surface helicity in terms of linkage of distinct field lines of the underlying vector field. In <Ref> we provide an alternative interpretation on toroidal surfaces which connects surface helicity to other important dynamical quantities on the torus. Based on this alternative interpretation we connect in <Ref> the notion of surface helicity to the notion of rotational transform in the context of plasma fusion. We then deal in <Ref> with the question of optimising surface helicity among toroidal surfaces of prescribed area and prove that toroidal surfaces admitting a symmetry are global minimisers while the existence of global maximisers is stated as an open problem. Finally, in <Ref> we discuss how some of the results obtained in the present manuscript can be used to construct "simple" surface currents on a coil winding surface which approximate desired target magnetic fields in a given plasma domain, which is of relevance in stellarator plasma fusion confinement devices. The proofs of the theorems are contained in <Ref>.



§ MAIN RESULTS

Before we state the main results we establish some notation.
Notation: For a given, C^1,1-regular, closed, connected surface Σ⊂ℝ^3 we let L^p(Σ), 1≤ p≤∞, denote the standard space of L^p-integrable functions on Σ and we define the vectorial counterpart by

    L^p𝒱(Σ):={v∈(L^p(Σ))^3| v(x)·𝒩(x)=0 a.e. x∈Σ},

where 𝒩 denotes the outward pointing unit normal on Σ, i.e. the unit normal pointing in the direction of the unbounded component of ℝ^3∖Σ. In addition we let C^0,1𝒱(Σ) denote the Lipschitz-continuous vector fields on Σ which are tangent to Σ at every point. We say that v∈ L^p𝒱(Σ) is divergence-free if ∫_Σ v·∇_Σα dσ=0 for all α∈ C^1(Σ), where ∇_Σ denotes the surface gradient of α which can be computed as ∇_Σα=∇α̃-(𝒩·∇α̃)𝒩, where α̃ is an arbitrary C^1-extension of α to Ω and ∇ denotes the standard Euclidean gradient, and where dσ denotes the standard surface measure. We denote by L^p𝒱_0(Σ) the subspaces of L^p𝒱(Σ) which consist of divergence-free vector fields. Lastly, we say that v∈ L^p𝒱(Σ) is curl-free if ∫_Σv· (∇_Σα×𝒩)dσ=0 for all α∈ C^1(Σ).


 §.§ Properties of Surface helicity

We start by defining the 2-dimensional analogue of the Biot-Savart operator.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Given 1<p<∞, the following operator is a well-defined, continuous operator called the surface Biot-Savart operator
	
    BS_Σ:L^p𝒱(Σ)→ L^q𝒱(Σ), v↦1/4π(∫_Σv(y)×x-y/|x-y|^3dσ(y))^∥

	where q:=2p/2-p if 1<p<2, where 1≤ q<∞ can be taken arbitrary if p=2 and q=∞ if 2<p≤∞.

In order to define the "tangent part" ·^∥ we cover Σ by any collection of open domains U_k which support local orthonormal frames {E^k_1,E^k_2} of class C^0,1. On each such U_k we define (we drop the superscript k)

    (∫_Σ v(y)×x-y/|x-y|^3dσ(y))^∥:=∑_i=1^2(∫_Σv(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x).

The well-definedness of <Ref> is shown in <Ref>. In particular we show that the defined vector fields agree on the overlaps of the U_k almost surely and that any two covers U_k and Ũ_k induce the same vector field.
We come now to the definition of (cross-)helicity.

	
	Let Σ⊂ℝ^3 be a connected, closed, C^1,1-surface. Then we define the cross-helicity as follows
	
    ℋ_c:L^4/3𝒱(Σ)× L^4/3𝒱(Σ)→ℝ, (v,w)↦∫_Σv(x)·BS_Σ(w)(x)dσ(x)
    		
    
    		=1/4π∫_Σ×Σv(x)·(w(y)×x-y/|x-y|^3)dσ(y)dσ(x).

Further, we define the surface helicity of a vector field by

    ℋ:L^4/3𝒱(Σ)→ℝ, v↦ℋ_c(v,v).



	
	Due to the continuity of BS_Σ:L^4/3𝒱(Σ)→ L^4𝒱(Σ) we see that cross helicity (and therefore helicity) is continuous and that there is some constant c(Σ)>0 such that for all v,w∈ L^4/3𝒱(Σ) we have |ℋ_c(v,w)|≤ c_0v_L^4/3(Σ)w_L^4/3(Σ).

In <cit.> Cantarella and Parsley introduced the notion of helicity of closed forms on (sub-)manifolds of Euclidean spaces. Of particular interest are the (1,3,3)-helicity and the (0,2,3)-helicity which correspond to helicities of 2-forms on domains in ℝ^3 and of 1-forms on embedded surfaces in ℝ^3 respectively. It is shown <cit.> that the (1,3,3)-helicity coincides with the classical helicity of divergence-free fields, tangent to the boundary of a given domain Ω upon an identification of a given closed 2-form α with a vector field v by contraction of the corresponding volume form.

Regarding the (0,2,3)-helicity the authors in <cit.> mention that it should be possible to regard it as a measure of the linkage of distinct field lines of a vector field associated with a given closed 1-form α and they raise the question of whether or not the (0,2,3)-helicity may attain non-trivial values (i.e. whether there exists v∈ C^∞𝒱_0(Σ) with ℋ(v)≠ 0). They conjecture that this should be the case whenever Σ has a non-trivial homology.

To summarise, the following two central problems regarding the (0,2,3) helicity were open <cit.>


	
  * Show that if Σ has non-trivial homology, then the (0,2,3)-helicity is not identically zero.
	
  * Provide a mathematically rigorous physical interpretation of (0,2,3)-helicity.

In the upcoming result we denote by ℋ_(0,2,3) the (0,2,3)-helicity on (closed) 1-forms and by ℋ the (vector field) helicity as introduced in <Ref>. Given some v∈ C^∞𝒱(Σ) we denote by ι_vω_Σ the contraction of the area form ω_Σ via the vector field v. In particular, if v is divergence-free, the associated 1-form is closed. Since the authors in <cit.> deal with smooth domains and surfaces we formulate the following result in the smooth setting.

	
	Let Σ⊂ℝ^3 be a smooth, connected, closed surface and let v∈ C^∞𝒱_0(Σ). Then
	
    ℋ_(0,2,3)(ι_vω_Σ)=ℋ(v).


We exploit this relationship to provide an answer to both of the open problems raised in <cit.> regarding (0,2,3)-helicity.

The following provides an answer to the first open problem regarding surface helicity.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then there exists some v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0 if and only if g(Σ)≥ 1, where g(Σ) denotes the genus of Σ. Further, if v∈ L^2𝒱_0(Σ) is co-exact, i.e. it can be written as v=grad_Σ(f)×𝒩, for some function f∈ H^1(Σ) and where 𝒩 denotes the outward unit normal, then ℋ_c(v,w)=0 for all w∈ L^2𝒱_0(Σ).

Regarding the second question posed above we intend to follow the ideas of <cit.> and <cit.> regarding the physical interpretation of the standard 3-d helicity (which corresponds to (1,3,3)-helicity in the notation of <cit.>).

Note that in general Σ might contain a large "flat region" (a copy of a large 2-d disc) and then if the field lines of a given vector field v∈ C^0,1𝒱_0(Σ) are not necessarily periodic, we might attempt, in the same spirit as in <cit.>, to close the field lines by geodesics. However, at least as long as the endpoints of the considered field lines are contained in the flat region, the geodesics will be lines passing through the corresponding endpoints of the distinct field line segments. But then these artificially added line segments will intersect with "high probability" (since the directions of the lines passing through the pairs of endpoints will be "generically" not parallel). Thus, one should expect that there is a set of positive measure in Σ×Σ for which the "artificially" closed field lines will intersect if we close field line segments by means of geodesics. In addition, the artificially added parts may intersect the original field line segments, see <Ref>.


Therefore it is necessary to find a new way to appropriately close the field lines of the underlying vector field to be able to obtain an interpretation of helicity in terms of average linking numbers.

Given some v∈ C^0,1𝒱_0(Σ) and x∈Σ we let γ_x(t) denote the (unique) field line of v starting at x. Now if we are given some x∈Σ and T>0 it can happen that γ_x(T)=x=γ_x(0) in which case the field line γ_x is closed (possibly a point). In general however we will find γ_x(T)≠ x=γ_x(0) and therefore it is necessary to close the field line in a suitable way to give an interpretation of helicity in terms of linking of suitable closed curves. It is well-known that, if Σ∈ C^2, there exists some τ_0(Σ)>0 such that the map Ψ_τ(x):=x+τ𝒩(x), x∈Σ, is a diffeomorphism onto its image for every |τ|<τ_0 where 𝒩 denotes the outward pointing unit normal, <cit.>. We define Σ_τ:=Ψ_τ(Σ) so that in particular Σ_0=Σ. Now it is the case that for almost every (x,y)∈Σ×Σ the images of the corresponding field lines γ_x(ℝ) and γ_y(ℝ) are disjoint. Given T,S>0 we then consider the curves γ_x:[0,T]→Σ and γ_y:[0,S]→Σ. In order to close them (in case they are not already closed) we proceed as follows: We connect γ_x(T) with Ψ_τ(γ_x(T)) for a suitably chosen τ (which we will be able to choose independent of x,y,T and S) by following the path Ψ_t(γ_x(T)) from t=0 until t=τ. Similarly, we can connect Ψ_τ(x) and x=γ_x(0) by inverting the path Ψ_t(x) from t=0 until t=τ. We note that Ψ_τ(γ_x(T)),Ψ_τ(x)∈Σ_τ and that we are left with connecting Ψ_τ(γ_x(T)) and Ψ_τ(x). We connect these two points by any curve contained in Σ_τ which is of unit speed and whose length can be uniformly bounded independently of x and T. For instance, we can connect these two points by any (length minimising) geodesic on Σ_τ. We close the curve γ_y:[0,S]→Σ in a similar fashion, but in contrast we connect y and γ_y(S) with their corresponding points on Σ_-τ and connect the corresponding two points on Σ_-τ by a geodesic on Σ_-τ. Let us denote these two closed curves obtained in that way by σ^τ_x,T and σ^-τ_y,S, see <Ref> for an illustration. By construction these two closed curves intersect if and only if γ_x([0,T]) and γ_y([0,S]) intersect, which we will show does not happen for almost every (x,y)∈Σ×Σ.
Note also that we explicitly leave the surface Σ to close our field lines.
By convention we set the linking number of a closed curve and any point not lying on this curve to zero.




With these preliminary observations we obtain the following result where we assume the surface Σ to be of class C^2 to avoid technicalities.

	
	Let Σ⊂ℝ^3 be a closed, connected C^2-surface. Then there exists some τ(Σ)>0 such that for every v∈ C^0,1𝒱_0(Σ) and every sequences (T_n)_n,(S_n)_n⊂ (0,∞) which are strictly increasing and diverging to ∞, the functions 
	
    Σ×Σ→ℝ, (x,y)↦lk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_n

	are well-defined elements of L^1(Σ×Σ) for every n and converge to a limit function in L^1(Σ×Σ). Further, we have the identities
	
    ℋ(v)=lim_n→∞∫_Σ×Σlk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_ndσ(x)dσ(y)=∫_Σ×Σlim^L^1_n→∞lk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_ndσ(x)dσ(y).


Here lim^L^1 indicates that we consider the L^1-limit (and not the pointwise limit which exists only upon passing to appropriate subsequences). Therefore surface helicity may be regarded as either the "asymptotic average linking number of distinct field lines of v" or, as is done more commonly in the 3-d case, the "average asymptotic linking number of distinct field lines of v".

	
	If all field lines of v are periodic and there exists some τ>0 such that all field lines of v have a period of at most τ, then we can instead close the field line segments of v by continuing to follow the flow of v until we return to the respective starting points. In this periodic situation no additional technical issues arise by assuming that Σ∈ C^1,1.



 §.§ Surface helicity on toroidal surfaces


We have seen in <Ref> that on an arbitrary (closed) surface Σ, we can interpret helicity as a measure of entanglement of distinct field lines of the underlying vector field. In the 3-dimensional situation entanglement of distinct field lines does not allow to infer information about the dynamical properties of individual field lines. However, in the 2-dimensional situation closed field lines are co-dimension 1 submanifolds and therefore the dynamics of individual field lines may very well influence the ability of the remaining field lines to link. To illustrate this point further, we imagine we are given a toroidal surface Σ≅ T^2 in ℝ^3 and a (regular) vector field v tangent to Σ. We suppose that v has a closed field line γ which wraps once in poloidal direction around Σ. Since distinct field lines do not cross we infer that the remaining field lines of v must be contained within Σ∖γ. But we can continuously "unfold" Σ∖γ and we see that in fact the remaining field lines may be viewed as being contained in a cylinder so that no linking may take place, c.f. <Ref>. Consequently, the helicity of v must be zero by means of the linking interpretation of helicity, <Ref>. We therefore see that the dynamics of individual field lines may have a strong influence on surface helicity and one may thus expect that an alternative interpretation of surface helicity in terms of averages of dynamical properties of individual field lines should be possible.




We focus here on C^1,1-surfaces Σ⊂ℝ^3 which bound solid tori, i.e. Σ=∂Ω for a domain Ω⊂ℝ^3 which is diffeomorphic to the solid torus D^2× S^1, where D^2 is the unit disc in ℝ^2 and S^1 is its boundary. We note that these type of surfaces are the most relevant in plasma physics, as they appear as the boundaries of plasma domains or model the surfaces along which the coils wind which generate the plasma confining magnetic field <cit.>.

Before we state the main result of this subsection let us introduce some notions.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3, i.e. Ω≅ D^2× S^1 and ∂Ω=Σ∈ C^1,1. We say that a simple closed C^1-curve σ_p on Σ is purely poloidal if σ_p is contractible when viewed as curve within Ω, but represents a non-trivial element of the first fundamental group of Σ. We call a simple closed C^1-curve σ_t in Σ toroidal if it generates the first fundamental group of Ω once viewed as a curve in Ω.

We note first that, up to orientation and a standard identification, any two purely poloidal curves are path-homotopic within Σ. Similarly, up to orientation, all toroidal curves are homotopic within Ω. However, two toroidal curves may not be (even up to orientation) homotopic within Σ. In addition, any choice of σ_p and σ_t forms a pair of generators of the first fundamental group of Σ.

To motivate the next definition we consider an arbitrary simple closed C^1-curve σ contained in Σ and for a fixed pair σ_p and σ_t of a purely poloidal and a toroidal curve we can express σ=Pσ_p⊕ Qσ_t where ⊕ denotes the concatenation of curves and P,Q∈ℤ denote the amount of times the curves σ_p and σ_t are traversed. We observe first that there is a unique pair γ_p,γ_t∈ℋ(Σ):={v∈ W^1,2𝒱(Σ)|curl_Σ(v)=0=div_Σ(v)} which forms a basis of ℋ(Σ) and satisfies

    ∫_σ_pγ_t=0=∫_σ_tγ_p, ∫_σ_pγ_p=1=∫_σ_tγ_t.

We observe further, which for example follows from an identical reasoning as in <cit.>, that in fact ℋ(Σ)⊂⋂_1<p≤∞W^1,p𝒱(Σ)⊂⋂_0<α<1C^0,α𝒱(Σ) and therefore the line integrals in (<ref>) make sense in a classical sense. Now since σ is homotopic to Pσ_p⊕ Qσ_t, we find

    ∫_σγ_t=P∫_σ_pγ_t+Q∫_σ_tγ_t=Q

and further

    lim_T→∞1/T∫_σ[0,T]γ_t=Q/τ,

where τ>0 is the period of σ and σ[0,T] is the path [0,T]→Σ, t↦σ(t) (which is not simple for T≥τ). Similarly we find P=∫_σγ_p and lim_T→∞1/T∫_σ[0,T]γ_p=P/τ. This motivates the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a toroidal curve σ_t and a purely poloidal curve σ_p we define the weighted asymptotic toroidal windings resp. weighted asymptotic poloidal windings of a (not necessarily periodic) C^1-curve σ:[0,∞)→Σ by
	
    q̂(σ):=lim_T→∞1/T∫_σ[0,T]γ_t, p̂(σ):=lim_T→∞1/T∫_σ[0,T]γ_p.



	
	We note that γ_t in (<ref>) is independent of the chosen purely poloidal curve σ_p and it only depends on the toroidal curve σ_t via its orientation, i.e. if two toroidal curves are oriented such that they are homotopic within Ω then the corresponding γ_t will be the same and if they are oriented in opposite direction, then the corresponding γ_t will differ by a minus sign. Consequently, q̂(σ) also depends only on the chosen orientation of the toroidal curve. However, γ_p will in general depend on the homotopy type of σ_t when viewed as a curve in Σ so that even if two toroidal curves are homotopic within Ω, the corresponding γ_p may be distinct (but will be the same if the toroidal curves are homotopic within Σ).
		
The following is the natural generalisation to vector fields.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a toroidal curve σ_t, a purely poloidal curve σ_p and a vector field v∈ C^0,1𝒱_0(Σ), we define the weighted asymptotic toroidal windings and the weighted asymptotic poloidal windings of v at a point x∈Σ as follows
	
    q̂(x):=q̂(γ_x), p̂(x):=p̂(γ_x),

	where γ_x denotes the field line of v starting at x. We further define the average weighted asymptotic toroidal windings and the average weighted asymptotic poloidal windings of v by
	
    Q(v):=∫_Σq̂(x)dσ(x)/|Σ|, P(v):=∫_Σp̂(x)dσ(x)/|Σ|,

	where |Σ| denotes the area of Σ.

We note that in order to compute the average one should have q̂,p̂∈ L^1(Σ). This however follows from standard ergodic theoretical arguments. We state it as a separate lemma since it is of importance when establishing a connection between surface helicity and toroidal and poloidal averages.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve respectively. If v∈ C^0,1𝒱_0(Σ), then q̂,p̂∈ L^1(Σ) and we have
	
    Q(v)=∫_Σv(x)·γ_t(x)dσ(x)/|Σ|, P(v)=∫_Σv(x)·γ_p(x)dσ(x)/|Σ|.



	One natural way to define purely toroidal simple closed curves is to demand that a given toroidal loop is trivial within ℝ^3∖Ω, see also <cit.>. Another useful way to choose a toroidal curve σ_t can consist in making sure that the resulting induced element γ_p is L^2(Σ)-orthogonal to γ_t so that the quantities Q(v) and P(v) essentially correspond to the L^2(Σ)-orthogonal projection of v onto the space ℋ(Σ) in the sense that π(v)=|Σ|(Q(v)γ_t/γ_t^2_L^2(Σ)+P(v)γ_p/γ_p^2_L^2(Σ)) where π denotes the projection onto ℋ(Σ) and v∈ C^0,1𝒱_0(Σ). Both choices can have advantages in distinct situations and therefore we allow general toroidal curves in our considerations.

To establish a connection between the quantities Q and P, which make statements about the average behaviour of individual field lines, and helicity which is a measure of the average linking of distinct field lines, we need to introduce one final notion.
We define the space ℋ_N(Ω) of harmonic Neumann fields on a bounded C^1,1-domain Ω⊂ℝ^3 as follows

    ℋ_N(Ω):={Γ∈ H^1(Ω,ℝ^3)|curl(Γ)=0=div(Γ), Γ∥∂Ω}.

We note first, c.f. <cit.>, that ℋ_N(Ω)⊂⋂_1≤ p<∞W^1,p𝒱(Ω) and thus, due to Sobolev embeddings ℋ_N(Ω)⊂⋂_0<α<1C^0,α𝒱(Ω). Since curl(Γ)=0 it follows that the restriction Γ|_Σ satisfies curl_Σ(Γ|_Σ)=0 in the weak sense and according to the Hodge-decomposition theorem <cit.> we can therefore write

    Γ|_Σ=∇_Σα+γ

for suitable α∈ H^1(Σ) and γ∈ℋ(Σ). Here γ is the L^2-orthogonal projection of Γ|_Σ onto ℋ(Σ). We note that since ℋ_N(Ω) is L^2(Ω)-orthogonal to the gradient fields and because Γ=curl(A) for a suitable vector potential A (for instance A may be taken to be the Biot-Savart potential of Γ) we must have γ≠ 0 whenever Γ≠ 0.

We are now finally ready to give an alternative interpretation of surface helicity in terms of the "dynamics of individual field lines".

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p be a purely poloidal curve and σ_t be a toroidal curve. Further, fix any Γ∈ℋ_N(Ω) such that the L^2(Σ)-orthogonal projection γ of Γ|_Σ onto ℋ(Σ) is L^2(Σ)-normalised. In addition let γ̃:=γ×𝒩. Then, given any v∈ C^0,1𝒱_0(Σ), we have
	
    ℋ(v)/|Σ|^2=Q(v)P(v)∫_σ_tγ∫_σ_pγ̃+[ℋ(γ)(∫_σ_tγ)^2+∫_σ_tγ∫_σ_tγ̃]Q^2(v).

	If, moreover, σ_t bounds a bounded C^1,1-surface 𝒜 outside of Ω, i.e. 𝒜⊂ℝ^3∖Ω and ∂𝒜=σ_t, then
	
    ℋ(v)/|Σ|^2=Q(v)P(v)∫_σ_tγ∫_σ_pγ̃.



	
	We observe that (ℋ_N(Ω))=1 whenever Ω is a solid torus, <cit.>. Hence the condition of L^2(Σ)-normalisation of γ determines Γ uniquely up to a minus sign. If a chosen Γ is replaced by -Γ we see that γ as well as γ̃ both are also changed by a minus sign and hence (since ℋ(-γ)=ℋ(γ)) the above formula is independent of that choice, as one would expect. Similarly, if σ_p and σ̃_p are purely poloidal curves on Σ then either they are homotopic and thus the integrals involving σ_p are unchanged or otherwise σ_p and σ̃_p are oppositely oriented, in which case ∫_σ̃_pγ̃=-∫_σ_pγ̃, but P(v) also changes sign (since it is defined via γ_p which in turn is defined via σ_p). In addition, γ_t is independent of the chosen σ_p and thus Q is also independent of that choice. Therefore the right hand side of (<ref>) is independent of the chosen purely poloidal curve. Finally, we observe that Γ|_Σ and γ differ only by a gradient field such that ∫_σ_tγ=∫_σ_tΓ from which it follows that ∫_σ_tγ only depends on the orientation of σ_t (considered as a curve in Ω) and it is clear once again that changing the orientation of σ_t leaves the right hand side of (<ref>) invariant. Since changing orientation of σ_t leaves the right hand side invariant, we are left with understanding how the quantities ∫_σ_tγ̃, P(v) and Q(v) change if we replace σ_t by a toroidal curve σ̃_t which is homotopic to σ_t in Ω. It follows easily from (<ref>) that γ_t and hence Q(v) remains unchanged. On the other hand, since σ_t and σ_p form a basis of the fundamental group of Σ, we may express σ̃_t=σ_t⊕ Pσ_p, where we used that we may assume that σ̃_t is homotopic to σ_t. Then ∫_σ̃_tγ̃=∫_σ_tγ̃+P∫_σ_pγ̃ and one can verify that γ̃_p-γ_p=-Pγ_t so that it follows from <Ref> that P(v)-P(v)=-PQ(v) (where γ̃_p, P(v) are computed with respect to σ_p and σ̃_t). We hence see that the additional terms appearing through the change of P(v) and ∫_σ_tγ̃ when σ_t is replaced by σ̃_t cancel each other. We conclude that the right hand side of (<ref>) is independent of the chosen curves σ_p and σ_t, as it should be because the left hand side of (<ref>) is of course independent of any such choices.



 §.§ Surface helicity and rotational transform


Assume once more that T^2≅Σ⊂ℝ^3 bounds a C^1,1-solid torus Ω⊂ℝ^3 and suppose that we fixed some purely poloidal curve σ_p and a toroidal curve σ_t on Σ. Now, if σ:ℝ→Σ is any other closed C^1-curve, then we can write it as σ=Pσ_p⊕ Qσ_t, where Q may be viewed as the "toroidal" turns of σ and P may be viewed as the "poloidal" turns of σ. We recall that according to (<ref>) we can write

    Q/τ=lim_T→∞1/T∫_σ[0,T]γ_t, P/τ=lim_T→∞∫_σ[0,T]γ_p,

where γ_p,γ_t are determined by the equations in (<ref>). Consequently, if we are interested in the ratio of "poloidal" twists by "toroidal" twists P/Q, we obtain the identity (assuming Q≠ 0)

    ι(σ):=P/Q=lim_T→∞1/T∫_σ[0,T]γ_p/lim_T→∞1/T∫_σ[0,T]γ_t=p̂(σ)/q̂(σ),

where p̂(σ),q̂(σ) denote the "asymptotic windings" of σ. These observations motivate the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve respectively and let v∈ C^0,1𝒱(Σ). If q̂(x)≠ 0 for a given x∈Σ, c.f. <Ref>, we define the rotational transform of v at x as
	
    ι_v(x):=p̂(x)/q̂(x),

provided the limits in the definition of p̂(x) and q̂(x) exist.


	
	
		
  * According to <Ref>, if v∈ C^0,1𝒱_0(Σ), then p̂(x) and q̂(x) exist for a.e. x∈Σ w.r.t. the standard surface measure.
		
  * We observe that the definition of the rotational transform depends on the choice of σ_p and σ_t in the following way: We recall first the defining equations (<ref>) of γ_p and γ_t
		
    ∫_σ_pγ_t=0=∫_σ_tγ_p, ∫_σ_pγ_p=1=∫_σ_tγ_t.

		If σ̃_p is any other closed, purely poloidal curve, then either σ̃_p is homotopic to σ_p or otherwise it is homotopic to -σ_p (the curve obtained from σ_p by inverting the orientation). In both cases γ_t remains unchanged, while γ_p changes sign and so q̂ remains unchanged, while p̂ changes sign, i.e. at most the sign of ι is changed, depending whether σ̃_p is oriented in the same way as σ_p or oppositely oriented. On the other hand, if we change σ_t to some σ̃_t, then by definition either σ̃_t will be homotopic to σ_t or -σ_t as a curve in Ω (but not necessarily within Σ). Since σ_p and σ_t generate the first fundamental group of Σ, we can then however express σ̃_t=±σ_t⊕ Pσ_p for a suitable P∈ℤ, where the sign in ± depends on whether σ̃_t is homotopic to σ_t or -σ_t within Ω. Consequently, we see that γ_t at most changes sign, while γ̃_p=γ_p∓ Pγ_t. Thus, q̂ may change its sign, depending whether σ_t and σ̃_t are oriented in the same, or opposite, way within Ω and p̂̃̂(x)=p̂(x)∓ Pq̂(x), where p̂̃̂ indicates that this quantity is computed with respect to the curve σ̃_t. In conclusion ι̃_v(x)=±ι_v(x)∓ P, where once more the tilde indicates that ι_v is computed with respect to the toroidal curve σ̃_t. In particular, if σ_p,σ̃_p are two purely poloidal curves of the same orientation and σ_t,σ̃_t are two poloidal curves of the same orientation (within Ω), then ι̃_v(x)=ι_v(x)+β for some constant β∈ℤ, which is independent of x and the vector field v under consideration. Hence, if we wish to study a problem which aims to maximise ι (possibly under some additional constraints), then the maximisers will be independent of the chosen curves (as long as they are oriented in the same way). Finally, also notice that the property q̂(x)≠ 0 at a given x∈Σ is independent of the chosen curves.
		

We will now first state the main mathematical result of this section, before we shortly discuss the relevance of rotational transform in the context of plasma fusion confinement devices.

In order to do so we state here the (standard) definition of rectifiability of a vector field on a 2-torus T^2=ℝ^2ℤ^2.

	
	Let T^2≅Σ be a C^1,1-surface, which is diffeomorphic to the 2-torus. A vector field v∈ C^0,1𝒱(Σ) is said to be rectifiable if there exists a diffeomorphisms ψ:Σ→ T^2 such that ψ_*v=a∂_1+b∂_2 for suitable constants a,b∈ℝ and the standard coordinate fields ∂_1,∂_2 on T^2, where ψ_*v denotes the corresponding pushforward vector field.

With this we can formulate the main result of this subsection.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve on Σ respectively. Further, let γ_p,γ_t be a basis of ℋ(Σ) determined by the condition (<ref>). Assume that v∈ C^0,1𝒱(Σ) is rectifiable and that there exists some f∈ C^0,1(Σ,(0,∞)) such that w:=fv∈ C^0,1𝒱_0(Σ), i.e. div_Σ(w)=0. If Q(w)≠ 0, then the rotational transforms of v and w are well-defined at a.e. x∈Σ and there holds
	
    ι_v(x)=ι_w(x)=∫_Σw·γ_pdσ/∫_Σw·γ_t dσ=P(w)/Q(w),

	where the subscript indicates whose rotational transform is considered. In particular, ι_v(x) is independent of x.


	
	In general, for a given vector field v∈ C^0,1𝒱(Σ), even if the rotational transform ι_v(x) is an element of L^1(Σ), it is not true that 1/|Σ|∫_Σι(x)dσ(x)=P(v)/Q(v) because the average of a quotient does not need to coincide with the quotient of the respective averages.

As an immediate consequence of <Ref> and <Ref>, we obtain the following relation between surface helicity and rotational transform.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p be a purely poloidal curve on Σ and let σ_t be a toroidal curve on Σ which bounds a bounded C^1,1-surface 𝒜 outside of Ω. Assume further that v∈ C^0,1𝒱(Σ) is rectifiable and that there exists some f∈ C^0,1(Σ,(0,∞)) such that w:=fv∈ C^0,1𝒱_0(Σ), i.e. div_Σ(w)=0. If Q(w)≠ 0, then
	
    ℋ(w)/|Σ|^2=ι_wQ^2(w)(∫_σ_tγ)(∫_σ_pγ̃),

	where ι_w is the constant rotational transform of w and γ,γ̃ are defined as in <Ref> and in particular are independent of w.


	
	
		
  * We point out shortly that a corresponding formula for the vector field v does not need to hold necessarily, since we require the divergence-free property in order to invoke <Ref>, which is the reason why the conclusion is solely formulated for the vector field w.
		
  * It is also customary to call a vector field v∈ C^0,1𝒱(Σ) semi-rectifiable, provided there exists a (strictly) positive C^0,1-function f such that fv is rectifiable, c.f. <cit.>. Hence <Ref> can be reformulated by assuming that a given w∈ C^0,1𝒱_0(Σ) is semi-rectifiable.
	


	
	In the realm of plasma physics one is interested in steady solutions of the equations of ideal magnetohydrodynamics with a resting plasma <cit.>. Such plasma equilibria are determined by the PDE
	
    B×curl(B)=∇ p and div(B)=0 in 𝒫, B∥∂𝒫

	where 𝒫 denotes the region containing the plasma ("plasma region"), B is the corresponding magnetic field and p is the pressure. According to Arnold's structure theorem, <cit.>, <cit.>, if B is analytic and B and curl(B) are not everywhere collinear, the domain 𝒫 will decompose, after removing a 2-dimensional analytic subset from it, into a finite number of chambers, each of which is invariant under the flow of B and is one of the following two types:
	
		
  * Either the chamber foliates further into toroidal surfaces, each of which is again invariant under the flow of B and such that on each such surface B is rectifiable and hence in particular either all field lines of B are closed (with the same period) or all of the field lines of B are dense within the surface.
		
  * Or the chamber foliates into cylindrical surfaces diffeomorphic to S^1×ℝ, each of which is invariant under the flow of B and such that all field lines of B are closed on any such surface.
	
	Of particular interest are the invariant toroidal surfaces. In particular, in order to counter the plasma particle drift in a fusion plasma it is desirable to have a high rotational transform <cit.>, see also <Ref> for a precise relationship between the different notions of rotational transform. We therefore wish to understand the quantity ι_B on regular toroidal level sets of the pressure p. We note that if Σ is a regular toroidal level set of p, then |∇ p|≠ 0 on all of Σ and in fact, letting f:=1/|∇ p| one can verify that B:=fB is a divergence-free field on Σ (divergence-free with respect to the surface measure on Σ), c.f. <cit.>. It follows therefore from <Ref> that we can compute the rotational transform of B on a given regular toroidal level set of Σ as
	
    ι_B(x)=∫_ΣB·γ_pdσ/∫_ΣB·γ_tdσ

	for a fixed purely poloidal curve σ_p and a toroidal curve σ_t bounding a surface outside of the domain enclosed by the level set and their corresponding induced harmonic fields γ_p,γ_t∈ℋ(Σ). Note once more, that the rotational transform of B (on any fixed regular toroidal surface) is constant.



 §.§ Surface helicity optimisation


In <cit.> Cantarella, DeTurck, Gluck and Teytel considered the following optimisation in 3-dimensional space: First, given a smooth, bounded domain Ω⊂ℝ^3, they show that there exists some smooth vector field v on Ω with div(v)=0, v∥∂Ω such that

    ℋ_3D(v)/v^2_L^2(Ω)=sup_w≠ 0
    div(w)=0
     w∥∂Ωℋ_3D(w)/w^2_L^2(Ω)=:λ(Ω),

where ℋ_3D denotes the standard 3-dimensional Biot-Savart helicity, see <cit.> for a proof of the above fact. Further, <cit.>, implies that |λ(Ω)|≤ c_0√(|Ω|) for an absolute constant c_0 independent of Ω. In view of this it is natural to ask whether there exists a bounded, smooth domain Ω which maximises λ(Ω) among all other domains of the same volume. Physically this amounts to finding a domain Ω which supports a magnetic field (tangent to the boundary of Ω) whose field lines are (on average) as entangled as possible among all other magnetic fields and all other domains of the same volume.

Some necessary conditions, assuming the existence of optimal domains, have been derived in <cit.>. Up to this day the question of existence of optimal domains for this problem has remained open, even though some recent advances have been made within certain classes of domains, c.f. <cit.>,<cit.>.
Here we consider a corresponding problem for the surface helicity. The first result is the following.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then there exists some v_±∈ L^2𝒱_0(Σ) with v_±_L^2(Σ)=1 and such that
	
    ±ℋ(v_±)=max_w≠ 0
    
    		w∈ L^2𝒱_0(Σ)±ℋ(w)/w^2_L^2(Σ)=:Λ_±(Σ).

	Further,
	
		
  * If Σ≅ S^2, then every non-zero element of L^2𝒱_0(Σ) realises the above Rayleigh quotients.
		
  * If Σ≇S^2, then v_±∈ℋ(Σ) (i.e. v_± are additionally curl-free).
	

To state the next result we denote by π the L^2-orthogonal projection from L^2𝒱(Σ) onto L^2𝒱_0(Σ).

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then the operator π∘BS_Σ:L^2𝒱_0(Σ)→ L^2𝒱_0(Σ) is compact and self-adjoint and admits a largest positive and smallest negative eigenvalue. Further, v∈ L^2𝒱_0(Σ) maximises helicity among all other fields w∈ L^2𝒱_0(Σ) of the same L^2-norm if and only if v is an eigenfield of π∘BS_Σ corresponding to the largest positive eigenvalue. A corresponding result holds for the helicity minimisation problem.


	In general, it is highly non-trivial to compute the corresponding eigenfields of π∘BS_Σ which realise the Rayleigh quotients in (<ref>) even for rotationally symmetric surfaces. However, we provide in <Ref> an, in comparison, easy way to compute these eigenfields as well as the corresponding values Λ_+(Σ), Λ_-(Σ) if the surface Σ is toroidal. This will allow us to obtain exact solutions in the case of rotationally symmetric surfaces.

We now deal with the question of optimising the following quantity, recall (<ref>),

    Λ(Σ):=max{Λ_+(Σ),Λ_-(Σ)}=max_w≠ 0
    
    			w∈ L^2𝒱_0(Σ)|ℋ(w)|/w^2_L^2(Σ)

among C^1,1-closed surfaces. Note that according to <Ref> we have Λ(Σ)≥ 0 with equality if and only if Σ≅ S^2. Hence, Σ is a global minimiser of Λ(Σ) if and only if Σ≅ S^2. Here, due to their relevance in plasma physics, we want to focus on surfaces Σ≅ T^2.

	
	Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω be the domain of finite volume bounded by Σ. Then
	
    Λ(Σ)≥1/2.

	Equality holds if and only if ℋ(γ)=0, where γ is the L^2-orthogonal projection of Γ|_Σ onto ℋ(Σ) and Γ is any fixed non-zero harmonic Neumann field of Ω, i.e. any H^1-regular, curl-free, div-free vector field on Ω which is tangent to ∂Ω=Σ.


	With some more effort it should be possible to extend the same lower bound to surfaces of arbitrary (strictly positive) genus using similar methods as in the proof of <Ref> by working with a suitable basis of ℋ_N(Ω) in order to obtain an appropriate matrix representation of the surface Biot-Savart operator. However, we do not pursue such a possible extension further and instead list this as an open problem below.

Using the linking interpretation of helicity, <Ref> and <Ref>, we will be able to conclude the following.

	
	Let Σ⊂ℝ^3 be a closed, connected, C^1,1-regular, axisymmetric torus. Then Λ(Σ)=1/2. Consequently
	
    Λ(Σ)=min_T^2≅Σ∈ C^1,1Λ(Σ).



	
	We note that in <Ref> and <Ref> we did not specify the area of Σ. As we shall see, the scaling properties of the functional Λ imply that minimising or maximising Λ among surfaces with prescribed area is equivalent to optimising Λ among surfaces without an area constraint.

We recall that the corresponding 3-d problem motivated to try to find a global maximiser of Λ(Σ) rather than a global minimiser. We pose this as a remaining open problem.
Open Problems:

	
  * Does there exist a constant 0<c<∞ such that Λ(Σ)≤ c for all T^2≅Σ which are C^1,1-regular?
	
  * If the answer to (i) is positive, does there exist some T^2≅Σ such that Λ(Σ)=max_T^2≅Σ∈ C^1,1Λ(Σ)?
	
  * Do the answers to (i) and (ii) change if we drop the assumption T^2≅Σ and instead allow Σ to have arbitrary genus?
	
  * Does <Ref> extend to surfaces of arbitrary genus, i.e. is it true that every closed, connected C^1,1-surface Σ⊂ℝ^3 of genus g≥ 1 satisfies Λ(Σ)≥1/2?



 §.§ Simple Surface currents


When building plasma fusion confinement devices one wishes to generate magnetic fields in order to confine the fusion plasma. In particular, in so called stellarator devices, one uses a complex coil structure to generate the fields <cit.>. One way to model these complex coil structures is the so called coil winding surface (CWS) model, <cit.>, where one assumes that the coils are infinitely thin and wind around a given closed surface. The induced current is then modelled as a vector field tangent to the CWS and is assumed to be divergence-free in view of Maxwell's equations. Given a C^1,1-regular CWS Σ⊂ℝ^3 and a divergence-free current j on the CWS, the corresponding magnetic field induced by j within the domain Ω enclosed by Σ is given by the Biot-Savart law

    BS(j)(x)=1/4π∫_Σj(y)×x-y/|x-y|^3dσ(y), x∈Ω.

Now, for a fixed current j, there exist in general further currents j^' which induce the same magnetic field within Ω. In fact, since BS is a linear operator, we must have that j-j^'∈Ker(BS). To be more precise, we may view BS as a bounded operator from L^2𝒱_0(Σ) into L^2𝒱(Ω) (the square integrable vector fields on Ω), c.f. <cit.>. It has been further shown, see <cit.> that (Ker(BS))=g(Σ), where g(Σ) denotes the genus of Σ. In our applications, the CWSs are toroidal, i.e. T^2≅Σ, and thus (Ker(BS))=1. This gives us some flexibility and hence we may look for currents, and consequently coil configurations, which have a "simple" form. To this end we make the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a purely poloidal curve σ_p and a toroidal curve σ_t we further let γ_t∈ℋ(Σ) denote the induced harmonic field. Given a current j∈ L^2𝒱_0(Σ) we define Q(j):=∫_Σj·γ_t dσ/|Σ| and say that j simple if Q(j)=0. 


	
	
		
  * In view of <Ref>, <Ref> coincides with <Ref> whenever j is of class C^0,1.
		
  * We have previously explained in <Ref> that γ_t is independent of the choice of σ_p and that changing σ_t may at most change γ_t to -γ_t and therefore the definition of a simple current is independent of the chosen curves σ_p and σ_t.
		
  * If j∈ C^0,1𝒱_0(Σ), then according to <Ref> we see that j is simple if and only if ∫_Σq̂(x)dσ(x)=0, or in other words if and only if, on average, the weighted asymptotic toroidal windings of the field lines of j are zero. Therefore, in an oversimplified manner of speaking, the field lines of a simple current should not form any "knotted" structures and be expected to tend to be either more poloidal or contractible in nature.
		
  * A word of caution: It might turn out that the field lines of j are all toroidal curves and that simply on average the corresponding field lines point as often in positive toroidal direction as they point in negative toroidal direction. As an example consider Σ⊂ℝ^3 to be a standard rotationally symmetric torus and let ϕ denote the toroidal coordinate and θ the poloidal coordinate. We may then define j:=sin(θ)e_ϕ, where e_ϕ denotes the standard normalised toroidal field. Then j∈ C^0,1𝒱_0(Σ) and Q(j)=0, as follows easily from <Ref>, and hence j is simple. However, j has only a toroidal component, see <Ref>.
	





Here is our main result regarding the existence of simple current configurations.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Then for every j∈ L^2𝒱_0(Σ) there exists a unique j^'∈ L^2𝒱_0(Σ) such that BS(j^')=BS(j) in Ω and j^' is simple. Further, if j∈ L^p𝒱_0(Σ) for some 2≤ p≤∞ or j∈ C^0,α𝒱_0(Σ) for some 0≤α<1, then so is j^', where C^0,0≡ C^0 denotes the continuous currents.

In the context of plasma fusion confinement devices one of the situations one faces is the following: We suppose we are given a CWS Σ and a plasma region P which is assumed to be precompact within the finite domain Ω bounded by the CWS and is thought of to contain the fusion plasma. Further, we assume we are given a square integrable, harmonic, target magnetic field B_T∈ L^2ℋ(P):={X∈ L^2(P,ℝ^3)|curl(X)=0=div(X)} which we wish to generate by a surface current in order to confine the fusion plasma. It is known, see <cit.>, that under reasonable assumptions, the image of BS, with domain L^2𝒱_0(Σ), is contained in and L^2(P)-dense in the space L^2ℋ(P). We therefore obtain the following important corollary from <Ref>, where the slightly technical assumptions in the upcoming results are depicted in <Ref>.





	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Assume further, that we are given another C^1,1-solid torus P which is contained in Ω and of positive distance to Σ and that there exists a C^2-disc D⊂Ω such that ∂ D⊂Σ is a purely poloidal C^1-curve in Σ and such that D∩ P is also a disc bounded by a purely poloidal C^1-curve contained in ∂ P. Then for every B_T∈ L^2ℋ(P) and every ϵ>0 there exists a current j∈ L^2𝒱_0(Σ) with Q(j)=0 and BS(j)-B_T_L^2(P)≤ϵ.


Our final result provides a minimisation procedure which allows one to obtain the desired approximating simple currents as minimisers of a specific "energy functional". The procedure is a modification  of a minimisation procedure which has already been used successfully in the past. We first recall the already established procedure, c.f. <cit.>. Given a positive parameter λ>0 one can consider the following minimisation problem

    C(λ;B_T):=inf_j∈ L^2𝒱_0(Σ)(BS(j)-B_T^2_L^2(P)+λj^2_L^2(Σ)).

It follows from standard variational techniques and the convexity of the functional involved that the infimum is in fact a minimum and that there exists a unique current j_λ∈ L^2𝒱_0(Σ) which realises C(λ;B_T). Using the density of the image of the Biot-Savart operator within L^2ℋ(P) it has been shown that BS(j_λ)-B_T_L^2(P)→ 0 as λ↘ 0, see <cit.>. In general, there is no reason to expect that the obtained currents j_λ satisfy Q(j_λ)=0. However, one can enforce this condition by observing that 

    L^2𝒱^Q=0_0(Σ):={j∈ L^2𝒱_0(Σ)|Q(j)=0}

is an L^2-closed subspace, and hence a Hilbert space in its own right, and by setting up a corresponding minimisation procedure

    C_0(λ;B_T):=inf_j∈ L^2𝒱^Q=0_0(Σ)(BS(j)-B_T^2_L^2(P)+λj^2_L^2(Σ)).

Just like in the case of C(λ;B_T) it follows from standard arguments that the infimum in C_0(λ;B_T) is actually achieved and that the global minimiser is unique. Our final result is the following

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Assume further that we are given another C^1,1-solid torus P which is contained in Ω and of positive distance to Σ and that there exists a C^2-disc D⊂Ω such that ∂ D⊂Σ is a purely poloidal C^1-curve in Σ and such that D∩ P is also a disc bounded by a purely poloidal C^1-curve contained in ∂ P. Given any B_T∈ L^2ℋ(P) we denote by j^0_λ the (unique) global minimiser realising C_0(λ;B_T) in (<ref>) for fixed λ>0. Then BS(j^0_λ)-B_T_L^2(P)→ 0 as λ↘ 0.



§ PROOFS OF MAIN RESULTS




 §.§ Proof of <Ref>

Before we come to the proof let us recall how the (0,2,3)-helicity is defined. Given a closed 1-form α on a surface Σ we can consider the inclusion ι: (Σ×Σ)∖{x=y}→Σ×Σ and the projections π_x:Σ×Σ→Σ,(x,y)↦ x and π_y:Σ×Σ→Σ,(x,y)↦ y. We define the Gauss map g:(Σ×Σ)∖{x=y}→ S^2, (x,y)↦x-y/|x-y| and we let ω^2_S^2 denote the standard normalised(!) area form on S^2. We can then define α_x:=(π_x)^#α, α_y:=(π_y)^#α (here f^# denotes the pullback via a map f). Now, strictly speaking, the space C_2(Σ):=(Σ×Σ)∖{x=y} is first compactified by means of the Fulton-MacPherson compactification, c.f. <cit.>, and it is shown that the forms α_x, α_y as well as the Gauss map extend smoothly to the compactification <cit.>. Further, the compactification has a manifold structure <cit.>. Denoting by C_2[Σ] the compactification and with a slight abuse of notation we can define the helicity as

    ℋ_(0,2,3)(α):=∫_C_2[Σ]α_x∧α_y∧ g^#ω^2_S^2.

We note that the space C_2[Σ] is 4-dimensional and so it makes sense to integrate a 4-form. The same reasoning as in <cit.> tells us that it is enough to prove that for all (x,y)∈ (Σ×Σ)∖{x≠ y} we have the identity

    α_x∧α_y∧ g^#ω^2_S^2=1/4πv(x)·(v(y)×x-y/|x-y|^3)ω_Σ(x)∧ω_Σ(y)

where ω_Σ is the area-form on Σ induced by the Euclidean metric and where v is the unique vector field on Σ satisfying ι_vω_Σ=α.

	We prove more generally that if α and β are two 1-forms on Σ, then
	
    β_x∧α_y∧ g^#ω^2_S^2=1/4πw(x)·(v(y)×x-y/|x-y|^3)ω_Σ(x)∧ω_Σ(y)

	where w is the vector field associated with β and v is the vector field associated with α by means of contracting the area form. We fix any x,y∈Σ with x≠ y and let μ,η be charts of Σ around x and y respectively. Additionally, we choose μ×η as a chart around (x,y)∈ (Σ×Σ)∖{x=y}. Further, we let p,q be any two points contained in the domains of μ and η respectively. We see that, if we denote by dμ^i,dη^j the induced covector bases at p,q∈Σ respectively and with a slight abuse of notation by dμ^i(p,q),dη^j(p,q) the induced basis at (p,q)∈ (Σ×Σ)∖{x=y}, then, if we express β(p)=β_i(p)dμ^i(p), we find 
	
    β_x(p,q)=((π_x)^#β)(p)=∑_i=1^2β_i(p)dμ^i(p,q).

	Similarly, α_y(p,q)=∑_j=1^2α_j(q)dη^j(p,q). As for the remaining term g^#ω^2_S^2 we recall that we have the identity
	
    ω^2_S^2=1/4πι^#(x dy∧ dz-ydx∧ dz+zdx∧ dy)

	where ι denotes the inclusion S^2↪ℝ^3. Hence, we may view the Gauss map g as a map into ℝ^3 rather than S^2 and are left with computing the pullback
	
    G^#(x dy∧ dz-ydx∧ dz+zdx∧ dy)(p,q)

	where G:(Σ×Σ)∖{x=y}→ℝ^3,(x,y)↦x-y/|x-y|. In order to find an appropriate expression for the coefficients of this 2-form we observe that
	
    G_*(p,q)∂_μ^i(p,q)=∂_μ^i(p)+(p-q)·∂_μ^i(p)/|p-q|^2(q-p)/|p-q|
    
    		G_*(p,q)∂_η^j(p,q)=-∂_η^j(q)+(p-q)·∂_η^j(q)/|p-q|^2(q-p)/|p-q|

	where the vectors tangent to Σ are viewed as vectors in ℝ^3. Defining ω(x,y,z):=xdy∧ dz-ydx∧ dz+z dx∧ dy, we observe that by definition of the pullback we have
	
    (G^#ω)(p,q)(∂_μ^1(p,q),∂_μ^2(p,q))=ω(p-q/|p-q|)(G_*(p,q)∂_μ^1(p,q),G_*(p,q)∂_μ^2(p,q)).

	Letting r̂:=p-q/|p-q|, ν:=∂_μ_1(p) and ξ:=∂_μ_2(p) we note that we need therefore to compute
	
    ω(r̂)(ν-(r̂·ν)r̂,ξ-(r̂·ξ)r̂)

	and an explicit computation yields
	
    ω(r̂)(ν-(r̂·ν)r̂,ξ-(r̂·ξ)r̂)=(ν×ξ)·r̂.

	Using this formula we can compute all the coefficient functions and eventually arrive at
	
    (G^#ω)(p,q)
    		
    
    		=p-q/4π|p-q|^3·(∂_μ^1(p)×∂_μ^2(p)dμ^1(p,q)∧ dμ^2(p,q)+∂_η^1(q)×∂_η^2(q)dη^1(p,q)∧ dη^2(p,q).
    		
    . -∑_i,j=1^2∂_μ^i(p)×∂_η^j(q)dμ^i(p,q)∧ dη^j(p,q))

	where we again view the tangent vectors at Σ as vectors in ℝ^3. We observe that the 2-form β_x∧α_y only contains basis vectors of the form dμ^i(p,q)∧ dη^j(p,q) so that all the dμ^1∧ dμ^2 and dη^1∧ dη^2 terms from G^#ω will vanish in the product β_x∧α_y∧ G^#ω because at least one of the covectors dμ^i or dη^j in these expressions must repeat, so that only the mixed terms are of relevance. We obtain
	
    β_x(p,q)∧α_y(p,q)∧ g^#ω^2_S^2(p,q)
    		
    
    		=q-p/4π|p-q|^3·∑_i,j,k,m=1^2β_k(p)α_m(p)(∂_μ^i(p)×∂_η^j(q))dμ^k(p,q)∧ dη^m(p,q)∧ dμ^i(p,q)∧ dη^j(p,q)
    		
    
    		=q-p/4π|p-q|^3·((β_2(p)∂_μ_1(p)-β_1(p)∂_μ_2(p))× (α_1(q)∂_η^2(q)-α_2(q)∂_η^1(q)))dμ^1∧ dμ^2∧ dη^1∧ dη^2.

	We recall that we started with two points x,y∈Σ with x≠ y and μ and η were any arbitrary charts centred around these two points. In particular, we may pick our charts such that g_ij(x)=δ_ij=g̃_ij(y) at these two points where g_ij and g̃_ij denote the metric tensor coefficients in the respective coordinate charts. Further, with the right choice of orientation on Σ we may assume that ∂_μ^1(x)×∂_μ^2(x)=𝒩(x) where 𝒩 denotes the outward pointing unit normal. Using the musical isomorphism we can then identify the forms α and β with vector fields X and Z respectively (note that this identification differs from identifying a 1-form by contracting the area-form). We can then expand (x-y) in the basis ∂_μ^1(x),∂_μ^2(x),𝒩(x) and using the properties of the chart we find
	
    (x-y)× (β_2(x)∂_μ^1(x)-β_1(x)∂_μ^2(x))=[(y-x)· Z(x)]𝒩(x)+[(x-y)·𝒩(x)]Z(x).

	Applying the cyclic properties of the inner product we find
	
    (x-y)· [(β_2(x)∂_μ^1(x)-β_1(x)∂_μ^2(x))× (α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))]
    		
    
    		=(α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))·([(y-x)· Z(x)]𝒩(x)+[(x-y)·𝒩(x)]Z(x)).

	Now we note that if a∈ℝ^3 is an arbitrary vector, then upon expanding it in the (orthonormal) basis {∂_η^1(y),∂_η^2(y),𝒩(y)}, we obtain
	
    𝒩(y)· [X(y)× a]=α_1(y)a^2-α_2(y)a^1=(α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))· a.

	We hence arrive at
	
    β_x(x,y)∧α_y(x,y)∧ g^#ω^2_S^2(x,y)
    		
    
    		=𝒩(y)/4π|x-y|^3·[[((y-x)· Z(x))𝒩(x)+((x-y)·𝒩(x))Z(x)]× X(y)]ω_Σ(x)∧ω_Σ(y)
    		
    
    		=X(y)×𝒩(y)/4π|x-y|^3· [((y-x)· Z(x))𝒩(x)+((x-y)·𝒩(x))Z(x)]ω_Σ(x)∧ω_Σ(y)
    		
    
    		=X(y)×𝒩(y)/4π|x-y|^3· [(Z(x)×𝒩(x))× (y-x)]ω_Σ(x)∧ω_Σ(y)

	where we used the vector-triple product rule in the last step. Lastly, we observe that the vector field associated with a 1-form α by contracting the area-form can be obtained from the vector field associated with the 1-form α via the musical isomorphism by taking the cross product with the outward pointing unit normal. So if we let v,w be the vector fields associated with the 1-forms α and β by contracting the area-form respectively, we arrive at
	
    β_x(x,y)∧α_y(x,y)∧ (g^#ω^2_S^2)(x,y)=v(y)·(w(x)×y-x/4π|x-y|^3)ω_Σ(x)∧ω_Σ(y).

	Consequently
	
    ∫_C_2[Σ]β_x∧α_y∧ g^#ω^2_S^2=∫_Σ v(y)·BS_Σ(w)(y)dσ(y)

	and hence ℋ_(0,2,3)(α)=ℋ(v).



 §.§ Proof of <Ref>

We recall that we want to prove that for a given closed, connected surface Σ⊂ℝ^3 there exists some v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0 if and only if g(Σ)≥ 1, where g(Σ) denotes the genus of Σ. In addition, we also need to show that ℋ_c(v,w)=0 for all w∈ L^2𝒱_0(Σ) whenever v is a co-exact vector field, where ℋ_c denotes the cross-helicity and ℋ denotes the helicity of vector fields.

	
	Step 1: We first prove that if g(Σ)≥ 1, then there exists some element v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0.
	We first observe that each closed surface Σ bounds a bounded domain Ω⊂ℝ^3 with ∂Ω=Σ. We then have the well-known relation (ℋ_N(Ω))=g(Σ), where ℋ_N(Ω) denotes the space of curl-free, div-free, H^1-vector fields on Ω which are tangent to the boundary ∂Ω and g(Σ) denotes the genus of Σ, <cit.>. In our situation g(Σ)≥ 1 and so we can fix some non-zero Γ∈ℋ_N(Ω). Since Γ is tangent to Σ we may view it as a vector field on Σ. The fact that Γ is curl-free on Ω implies that so is its restriction to Σ (in the weak sense). We recall that ℋ_N(Ω)⊂⋂_0<α<1C^0,α𝒱(Ω) and hence we can write according to the Hodge decomposition theorem Γ|_Σ=grad_Σ(f)+γ for suitable f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ) where ℋ(Σ) denotes the space of curl-free and div-free fields on Σ of class ⋂_1≤ p<∞W^1,p. Since Γ is non-zero so must be γ (since if the restriction of a harmonic Neumann field to the boundary is a gradient field, then the harmonic Neumann field is already identically zero). We can now let 𝒩 denote the outward pointing unit normal and define γ̃:=γ×𝒩 (so that in turn γ=𝒩×γ̃). This new field γ̃ is still divergence-free and curl-free (in the language of differential forms the 1-form associated with γ̃ by the musical isomorphism can be obtained from that of γ by applying the Hodge-star operator). If either ℋ(γ)≠ 0 or ℋ(γ̃)≠ 0, then the theorem is proven (since both vector fields are div-free). If on the other hand ℋ(γ)=0=ℋ(γ̃), then we can consider the vector field v:=γ̃+γ and notice that, due to the symmetry property of the Biot-Savart operator, we have
	
    ℋ(v)=ℋ(γ)+ℋ(γ̃)+2ℋ_c(γ̃,γ)=2ℋ_c(γ̃,γ)

	where we recall that ℋ_c(γ̃,γ)=(γ̃,BS_Σ(γ))_L^2(Σ) is the "cross-helicity". We note that since  γ̃=γ×𝒩 we get by means of the cyclic properties of the Euclidean product
	
    4πℋ_c(γ̃,γ)=∫_Σ∫_Σ(γ̃(x)×γ(y))·x-y/|x-y|^3dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ((γ(x)×𝒩(x))×γ(y))·x-y/|x-y|^3dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ [(γ(x)·γ(y))𝒩(x)-(γ(y)·𝒩(x))γ(x)]·x-y/|x-y|^3dσ(y)dσ(x)
    		
    
    		=∫_Σ∫_Σγ(y)·[(𝒩(x)·x-y/|x-y|^3)γ(x)-(γ(x)·x-y/|x-y|^3)𝒩(x)]dσ(y)dσ(x).

	We first claim that
	
    ℝ^3→ℝ, y↦∫_Σ(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩̃(y))dσ(x)

	is a continuous function, where 𝒩̃ is any C^0,1-regular (compactly supported) extension of 𝒩 to ℝ^3. But this follows easily from the generalised dominated convergence theorem because we have the estimate
	
    |(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩̃(y))|≤c/|x-y|

	for a suitable constant c>0 which is independent of y and from the fact that the function ℝ^3→ℝ, y↦∫_Σ1/|x-y|dσ(x) is continuous.
	
	Therefore, if for some fixed y∈Σ we let (y_n)_n⊂Ω be any sequence converging to y, we find
	
    γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)𝒩(x)dσ(x)
    		
    
    		=γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩(y))dσ(x)
    		
    
    		=lim_n→∞γ(y)·∫_Σ(γ(x)·x-y_n/|x-y_n|^3)(𝒩(x)-𝒩̃(y_n))dσ(x)

	where we used that γ(y)·𝒩(y)=0 because γ is a tangent field of Σ. We observe that since y_n∉Σ and since 𝒩̃(y_n) is independent of x, the term involving 𝒩̃(y_n) vanishes because
	
    ∫_Σγ(x)·x-y_n/|x-y_n|^3dσ(x)=-∫_Σγ(x)·∇^Σ_x(1/|x-y_n|)dσ(x)=0 for all n

	where we used that γ is divergence-free and that x↦1/|x-y_n| defines a C^1-function on Σ for all fixed n because each y_n has a positive distance to Σ. We conclude
	
    γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)𝒩(x)dσ(x)=lim_n→∞γ(y)·∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x).

	On the other hand we have the well-known jump formula, <cit.> which tells us that we can find a suitable sequence (y_n)_n⊂Ω converging to y such that
	
    ∫_Σ(𝒩(x)·x-y/|x-y|^3)γ(x)dσ(x)=-4π/2γ(y)+lim_n→∞∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x).

	Combining our considerations so far, we see that for all y∈Σ there exists some sequence (y_n)_n⊂Ω converging to y such that
	
    4πℋ_c(γ,γ̃)=-4π/2γ^2_L^2(Σ)
    
    		+∫_Σlim_n→∞γ(y)·(∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x)-∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x))dσ(y).

	We observe now that γ admits a curl- and div-free extension to Ω, i.e. there exists some ṽ∈⋂_1≤ p<∞W^1,p𝒱(Ω) with curl(ṽ)=0, div(ṽ)=0 in Ω and ṽ^∥=γ, where ṽ^∥:=ṽ-(𝒩·ṽ)𝒩. Indeed, we recall that Γ|_Σ=grad_Σ(f)+γ for a suitable function f∈⋂_0<α<1C^1,α(Σ), and thus if we let f̃ denote the harmonic extension of f, c.f. <cit.>, then Γ-grad(f̃)∈⋂_1≤ p<∞W^1,p𝒱(Ω) is the desired curl- and div-free extension. One can then use Gauss' theorem and the fact that y_n∈Ω for fixed n, to obtain
	
    ∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x)-∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x)
    		
    
    		=∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)ṽ(x)dσ(x)-∫_Σ(ṽ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x)
    		
    
    		=4πṽ(y_n)+∫_Ωdiv(ṽ)(x)x-y_n/|x-y_n|^3d^3x+∫_Ωy_n-x/|y_n-x|^3×curl(ṽ)(x)d^3x=4πṽ(y_n),

	where the first identity follows from the fact that the terms involving the normal part of ṽ cancel each other. Taking the limit n→∞ we conclude
	
    4πℋ_c(γ,γ̃)=4π/2γ^2_L^2(Σ).

	We recall that ℋ(γ+γ̃)=2ℋ_c(γ,γ̃) and therefore we arrive at
	
    ℋ(γ+γ̃)=2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)>0

	because γ≢0. Hence, in any case, there must exist some γ̂∈ℋ(Σ) satisfying ℋ(γ̂)≠ 0.
	Step 2: In this step we prove that if w∈ L^2𝒱_0(Σ) and f∈ H^1(Σ), then ℋ_c(∇^⊥ f,w)=0, where ∇^⊥ f:=∇_Σf×𝒩. In particular, as ℋ(∇^⊥ f)=ℋ_c(∇^⊥ f,∇^⊥ f) we will get ℋ(∇^⊥ f)=0. Since on a genus zero surface all divergence-free fields are co-exact this will conclude the proof.
	First we observe that by a density argument we may without loss of generality assume that w,∇_Σf∈⋂_1≤ p<∞W^1,p𝒱(Σ). We follow the arguments of the first step. We simply replace γ(y) by w(y) and γ(x) by ∇_Σ f(x) in the arguments. The only step where the arguments need to be modified is in (<ref>). More specifically, the argument that γ(y)·𝒩̃(y_n) ∫_Σ(γ(x)·x-y_n/|x-y_n|^3)dσ(x) vanishes no longer applies because in our new situation γ(x) has to be replaced by ∇_Σ f(x) which is no longer divergence-free. However, we see that
	
    ∫_Σ(∇_Σf(x)·x-y_n/|x-y_n|^3)dσ(x)=-∫_Σ∇_Σ f(x)·∇^Σ_x1/|x-y_n|dσ(x)=∫_ΣΔ_Σf(x)/|x-y_n|dσ(x)

	where Δ_Σ:=div_Σ∘∇_Σ denotes the Laplace operator on Σ. We note that the function ℝ^3→ℝ, y↦∫_Σh(x)/|x-y|dσ(x) is continuous for any function h∈ L^p(Σ), for some p>2, and therefore
	
    lim_n→∞w(y)·𝒩̃(y_n)∫_Σ(∇_Σ f(x)·x-y_n/|x-y_n|^3)dσ(x)=w(y)·𝒩(y)∫_ΣΔ_Σ f(x)/|x-y|dσ(x)=0

	because 𝒩̃(y)=𝒩(y) (since 𝒩̃ is an extension of 𝒩) and because w(y) is tangent to Σ. The remaining arguments apply verbatim which leads us to
	
    2ℋ_c(∇^⊥ f,w)=( ∇_Σ f,w)_L^2(Σ)=0

	because w is divergence-free.



 §.§ Proof of <Ref>


	We recall that v∈ C^0,1𝒱_0(Σ) is a div-free Lipschitz vector field on Σ. Further, we denote by γ_x the (unique) integral curve of v starting at x and for given T>0 we denote by γ_x[0,T] the path along the field line γ_x from time t=0 until t=T.
	Step 1: In the first step we prove that for every T,S∈ (0,∞) the two path γ_x[0,T] and γ_y[0,S] are disjoint almost surely (and hence by construction the artificially closed path σ^τ_x,T and σ^-τ_y,S will be also almost surely disjoint and have a well-defined linking number).
	
	For given T,S>0 we consider the set B_T,S:={(x,y)∈Σ×Σ|γ_x[0,T]∩γ_y[0,S]≠∅}. Now we note that (x,y)∈ B_T,S⇔∃ z∈γ_x[0,T]∩γ_y[0,S].	
	
	In conclusion we must have γ_x(τ)=z=γ_y(λ)⇔ y=γ_x(τ-λ) for suitable 0≤τ≤ T, 0≤λ≤ S, where we used the properties of the flow. Hence, for any T,S>0 we have
	
    B_T,S⊂⋃_n∈ℕ{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}.

	We observe that the right hand side is independent of T and S and that by sigma-subadditivity it is enough to prove that {(x,y)∈Σ×Σ|y∈γ_x(-n,n)} is a null-set in order to show that for almost every (x,y)∈Σ×Σ the sets γ_x[0,T] and γ_y[0,S] do not intersect (in fact this null set is independent of the choice of T and S).
	
	Denoting by μ the Riemannian measure on Σ we find, using Fubini's theorem,
	
    (μ×μ)({(x,y)∈Σ×Σ|y∈γ_x(-n,n)})=∫_Σ∫_Σχ_{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}dσ(y)dσ(x).

	We observe that
	
    χ_{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}(x,y)=χ_B_x(y)

	where B_x:={y∈Σ|y∈γ_x(-n,n)} so that
	
    ∫_Σχ_B_x(y)dσ(y)=μ(B_x)=μ(γ_x(-n,n))=0

	where we used in the last step that γ_x(-n,n) is either a point or otherwise a local embedding of a 1-manifold and hence has Hausdorff-dimension at most 1.
	Step 2: In this step we prove the remaining claims of the theorem.
	In order to specify our choice of τ(Σ)>0, we observe first that there exists some c=c(Σ)>0 such that, c.f. <cit.>,
		
    |𝒩(x)· (x-y)|≤ c|x-y|^2 for all x,y∈Σ.

		Further, we note that the function
		
    κ:Σ×Σ→ℝ, (x,y)↦𝒩(x)·𝒩(y)

		is uniformly continuous and thus
		
    𝒩(x)·𝒩(y)≥1/2 for all x,y∈Σ, |x-y|<δ(Σ)

		for a suitable δ(Σ)>0. We can then choose τ(Σ)>0 such that τ(Σ)≤min{δ/3,1/6c} and such that Ψ_t(x):=x+t𝒩(x) is a diffeomorphism from Σ onto its image for every |t|≤τ.
		
		In order to obtain the desired linking interpretation we observe first that, c.f. <cit.>:
		
		For every f∈ L^1(Σ×Σ), the sequence of functions ∫_0^S_n∫_0^T_nf(γ_x(t),γ_y(s))dtds/T_nS_n admits a well-defined L^1-limit denoted by f̂∈ L^1(Σ×Σ) and 
    ∫_Σ×Σf(x,y)dσ(x)dσ(y)=∫_Σ×Σf̂(x,y)dσ(x)dσ(y).

		We define f(x,y):=1/4π(v(x)× v(y))·x-y/|x-y|^3∈ L^1(Σ×Σ), which can be seen by writing v(x)× v(y)=v(x)× (v(y)-v(x)), using the Lipschitz property of v and the fact that 1/|x-y| is integrable over Σ×Σ. We therefore arrive at
		
    ℋ(v)=1/4π∫_Σ×Σlim^L^1_n→∞∫_0^T_n∫_0^S_n(γ̇_x(t)×γ̇_y(s))·γ_x(t)-γ_y(s)/|γ_x(t)-γ_y(s)|^3dtds/T_nS_ndσ(x)dσ(y).

		We recall the linking integral for two piecewise differentiable closed curves σ_1,σ_2 with σ_i(0)=σ_i(T_i)
		
    lk(σ_1,σ_2)=1/4π∫_0^T_1∫_0^T_2(σ̇_1(t)×σ̇_2(s))·σ_1(t)-σ_2(s)/|σ_1(t)-σ_2(s)|^3dtds.

		Now, as discussed before the statement of <Ref>, we close the curves γ_x([0,T_n]) and γ_y([0,S_n]) and obtain closed curves σ^τ_x,T_n and σ^-τ_y,S_n. For notational simplicity we drop the index n in T_n and S_n. According to (<ref>) and (<ref>) we have to show that the additional terms involved in the expression of lk(σ^τ_x,T,σ^-τ_y,S) converge to zero in the L^1(Σ×Σ)-sense.
		
		To this end we first let a∈Σ∖γ_x(ℝ), 0≤ s≤τ be any fixed elements and we observe that
		
    |a-s𝒩(a)-γ_x(t)|^2=|a-γ_x(t)|^2+s^2-2s(a-γ_x(t))·𝒩(a).

		Now, since a,γ_x(t)∈Σ we can use the estimate |(a-γ_x(t))·𝒩(a)|≤ c|a-γ_x(t)|^2 so that
		
    |a-s𝒩(a)-γ_x(t)|^2≥ |a-γ_x(t)|^2+s^2-2cs|a-γ_x(t)|^2
    			
    ≥ |a-γ_x(t)|^2+s^2/2-2c^2|a-γ_x(t)|^4

		where we used the elementary inequality 2db≤d^2/ϵ+ϵ b^2 for all d,b∈ℝ, ϵ>0 with d=s, b=c|a-γ_x(t)|^2 and ϵ=2. Now we distinguish two cases: Either 2c^2|a-γ_x(t)|^2≤1/2⇔ |a-γ_x(t)|^2≤1/4c^2, in which case we obtain
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2+s^2/2≥|a-γ_x(t)|^2+s^2/4,

		otherwise we find |a-γ_x(t)|^2≥1/4c^2. But in this case we can use the estimate |2s(a-γ_x(t))·𝒩(a)|≤ 2s^2+|a-γ_x(t)|^2/2 and therefore in this case
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2/2-s^2≥|a-γ_x(t)|^2/4+1/16c^2-s^2.

		To estimate the last term we recall that we are interested only in the situation 0≤ s≤τ and thus we can use the fact that τ≤1/6c, i.e. 1/36c^2≥τ^2 and therefore 1/16c^2≥ 2τ^2≥ 2s^2 which yields
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2+s^2/4

		in this case. We conclude that in any case we can estimate a part of the linking integral as follows:
		
    |∫_0^T∫_0^τ(v(γ_x(t))×𝒩(a))·a-s𝒩(a)-γ_x(t)/|a-s𝒩(a)-γ_x(t)|^3dsdt|≤ 4c_v∫_0^T∫_0^τ1/|a-γ_x(t)|^2+s^2dsdt

		for a suitable constant c_v>0 which is given by the (finite) C^0(Σ) norm of v. We can further explicitly compute and estimate
		
    ∫_0^τ1/|a-γ_x(t)|^2+s^2ds=arctan(τ/|a-γ_x(t)|)/|a-γ_x(t)|≤π/2|a-γ_x(t)|.

		We recall that in our application we want to connect either the point y or γ_y(S) with the corresponding points on Σ_-τ via a straight line following the normal direction of the starting point. We note that γ_y(S)=ψ_S(y) where ψ denotes the flow of v and because v is divergence-free, the corresponding flow is area-preserving. Hence we want to consider the situation where a=ψ(y) for a suitable area preserving diffeomorphism of Σ. But then we can estimate
		
    ∫_Σ×Σ|∫_0^T∫_0^τ(v(γ_x(t))×𝒩(ψ(y)))·ψ(y)-s𝒩(ψ(y))-γ_x(t)/|ψ(y)-s𝒩(ψ(y))-γ_x(t)|^3dsdt|dσ(x)dσ(y)
    			
    ≤ 2π c_v∫_Σ×Σ∫_0^T1/|ψ(y)-γ_x(t)|dtdσ(x)dσ(y)=2π c_vT∫_Σ×Σ1/|x-y|dσ(x)dσ(y)

		where we used Fubini's theorem in the last step and the fact that ψ as well as γ_x(t)=ψ_t(x) are area preserving diffeomorphisms for every t. Therefore, if we divide the above expression by T and S and take the limit T,S→∞ we see that the corresponding part of the linking integral converges to zero in the L^1-sense.
		
		Now we connect Ψ_-τ(y) and Ψ_-τ(γ_y(S)) on the surface Σ_-τ by a curve γ which is of unit speed and whose length is uniformly bounded independently of y and S, for instance we can connect them by a length minimising geodesic on Σ_-τ in which case its length is always bounded by the intrinsic diameter of Σ_-τ. In that case we can use the fact that Σ_-τ has a positive distance to Σ and therefore |γ_x(t)-γ(s)|≥ d(-τ)>0 for a suitable d which is independent of x,t,y,S and s. Consequently one can use the rough, pointwise, upper bound
		
    |v(γ_x(t))×γ̇(s)·γ_x(t)-γ(s)/|γ_x(t)-γ(s)|^3|≤c_v/|γ_x(t)-γ(s)|^2≤c_v/d^2

		in order to see that the corresponding part of the linking integral converges to zero in the L^1-sense.
		The only remaining part of the linking integral which is not covered by the considerations made so far is of the following type
		
    ∫_0^τ∫_0^τ𝒩(b)×𝒩(a)·b+t𝒩(b)-a+s𝒩(a)/|b+t𝒩(b)-a+s𝒩(a)|^3dsdt

		where in our application a∈{y,γ_y(S)} and b∈{x,γ_x(T)}. But here we can argue similar in spirit as in the first case, namely we can consider
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2=|(b-a)+t𝒩(b)+s𝒩(a)|^2
    			
    
    			=|b-a|^2+|s𝒩(a)+t𝒩(b)|^2+2s𝒩(a)· (b-a)+2t𝒩(b)· (b-a).

		We note that, since a,b∈Σ and 0≤ t≤τ, |2t𝒩(b)· (b-a)|≤ 2cτ |b-a|^2. We recall that τ≤1/6c and thus |2t𝒩(b)· (b-a)|≤|b-a|^2/3 and so we arrive at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2/3+|s𝒩(a)+t𝒩(b)|^2=|b-a|^2/3+t^2+s^2+2st𝒩(a)·𝒩(b).

		We recall that if |a-b|≤δ(Σ) then 𝒩(a)·𝒩(b)≥1/2 and hence, since s,t≥ 0 we arrive in this case at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|a-b|^2/3+t^2+s^2≥|a-b|^2+t^2+s^2/6≥|a-b|^2+t^2/6.

		On the other hand, if |a-b|≥δ, we can estimate 2st≤t^2/2+2s^2 and thus
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2/3+t^2/2-s^2.

		Further, |b-a|^2/6≥δ^2/6≥τ^2≥ s^2 for all 0≤ s≤τ because by choice of τ we have τ≤δ/3. Consequently we arrive at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2+t^2/6 for all a,b∈Σ.

		From here on out we can argue identically as in the first part of the proof that the corresponding part of the linking integral converges to zero in the L^1-sense. Hence, combining our findings with (<ref>) and the definition of the linking integral (<ref>) we conclude the validity of the theorem.



 §.§ Proof of <Ref>

We start by proving <Ref>.

	We will prove the statement only for Q(v), since an identical reasoning applies to P(v). We recall first that for given v∈ C^0,1𝒱_0(Σ) we denote its integral curves starting at a point x∈Σ by γ_x. Further we have by definition q̂(x)=lim_T→∞1/T∫_γ_x[0,T]γ_t where γ_t∈ℋ(Σ) is a harmonic field on Σ defined by the relations ∫_σ_pγ_t=0 and ∫_σ_tγ_t=1, where σ_p and σ_t are some fixed purely poloidal and a toroidal closed curves respectively. Finally, if q̂∈ L^1(Σ), we had set Q(v):=∫_Σq̂(x)dσ(x)/|Σ|.
	To see that q̂∈ L^1(Σ) and Q(v)=∫_Σv(x)·γ_t(x)dσ(x)/|Σ| we spell out the definition of the line integral, which leads us to
	
    ∫_γ_x[0,T]γ_t=∫_0^T γ̇_x(τ)·γ_t(γ_x(τ))dτ=∫_0^Tv(γ_x(τ))·γ_t(γ_x(τ))dτ.

	Now we define the function f:Σ→ℝ, f(x):=v(x)·γ_t(x)∈ L^1(Σ) (recall that γ_t∈ L^2𝒱(Σ)). We note that γ_x(τ)=ψ_τ(x) where ψ_τ denotes the area-preserving flow of v. Hence, standard ergodic theoretical results, c.f. <cit.>, imply that q̂∈ L^1(Σ) and that Q(v)=∫_Σf(x)dσ(x)/|Σ|=∫_Σv(x)·γ_t(x)dσ(x)/|Σ| as was to be shown.

Before we come to the proof of <Ref> we will need one additional lemma. To this end we recall that ℋ_N(Ω) denotes the space of curl-free, div-free, vector fields tangent to the boundary of a given domain Ω.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω⊂ℝ^3 be the bounded domain enclosed by Σ, ∂Ω=Σ. Given any Γ∈ℋ_N(Ω) we let γ∈ℋ(Σ) denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto the space of harmonic fields ℋ(Σ). Then ℋ(γ×𝒩)=0, where 𝒩 denotes the outward pointing unit normal.


	We follow the proof of <Ref>. Let us set γ̃:=γ×𝒩. Then
	
    4πℋ(γ̃)=∫_Σ∫_Σγ̃(x)·(γ̃(y)×x-y/|x-y|^3)dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ(γ̃(x)×γ̃(y))·x-y/|x-y|^3dσ(x)dσ(y)=∫_Σ∫_Σ((γ(x)×𝒩(x))×γ̃(y))·x-y/|x-y|^3dσ(y)dσ(x).

	From here on all the arguments in the computations of 4πℋ_c(γ,γ̃) apply verbatim, with the only caveat that γ(y) has to be replaced by γ̃(y) in the appropriate places during the computations. This leads us to the identity 2ℋ(γ̃)=∫_Σγ̃(y)·γ(y)dσ(y)=0 because γ̃ and γ are pointwise everywhere orthogonal to each other.

For potential future reference we also state the following immediate consequence.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω⊂ℝ^3 be the bounded domain enclosed by Σ, ∂Ω=Σ. Given any Γ∈ℋ_N(Ω) we have Γ×𝒩∈ L^2𝒱_0(Σ) and ℋ(Γ×𝒩)=0.


	As mentioned in the proof of <Ref> we can decompose Γ|_Σ as Γ|_Σ=∇_Σf+γ for a suitable function f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ). Since taking the cross product with 𝒩 corresponds, in the language of differential forms, to applying the Hodge star operator, it is immediate that Γ×𝒩∈ L^2𝒱_0(Σ). In addition, Γ×𝒩 and γ×𝒩 differ only by a co-exact vector field, so that the corollary follows from <Ref> and <Ref>.


	Since by assumption T^2≅Σ we know from standard Hodge-theory that (ℋ(Σ))=2. Further, if we fix any Γ∈ℋ_N(Ω)∖{0} where Ω⊂ℝ^3 is the solid torus enclosed by Σ, we see that its L^2-orthogonal projection γ onto ℋ(Σ) on Σ defines a non-zero element of ℋ(Σ). Setting γ̃:=γ×𝒩 we note that γ̃∈ℋ(Σ) because taking the cross product with the outer unit-normal corresponds to applying the Hodge star operator to the corresponding 1-forms (obtained from the musical isomorphism). Now, if v∈ C^0,1𝒱_0(Σ) is an arbitrary vector field, we can decompose v by means of the Hodge decomposition theorem as v=∇_Σf×𝒩+αγ+βγ̃ for a suitable C^1-function f∈ C^1(Σ) and suitable constants α,β∈ℝ. We observe that γ and γ̃ have the same L^2-norms, so that γ̃ is L^2-normalised if γ is. Assuming that γ is L^2-normalised and keeping in mind that the Hodge decomposition is L^2-orthogonal we find
	
    α=∫_Σγ· vdσ(x), β=∫_Σγ̃(x)· v(x)dσ(x).

	Utilising <Ref> we conclude
	
    ℋ(v)=ℋ(αγ+βγ̃)=α^2ℋ(γ)+2αβℋ_c(γ,γ̃)+β^2ℋ(γ̃).

	According to <Ref> we have ℋ(γ̃)=0. In addition, we have shown in the proof of <Ref> that 2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)=1 due to the normalisation of γ. We arrive at
	
    ℋ(v)=α^2ℋ(γ)+αβ.

	Now we recall that we are given a purely poloidal curve σ_p and a toroidal curve σ_t which uniquely determine elements γ_p,γ_t∈ℋ(Σ) according to the relations ∫_σ_pγ_t=0=∫_σ_tγ_p and ∫_σ_pγ_p=1=∫_σ_tγ_t. We claim that γ_t=γ/∫_σ_tγ. To see this we use the fact that Γ|_Σ-γ is the gradient of a C^1-function so that ∫_σ_pγ=∫_σ_pΓ. Further, we may assume that the C^1-curve σ_p, being purely poloidal, bounds a C^1 disc D⊂Ω, ∂ D=σ_p. The idea now is to apply Stokes' theorem to deduce ∫_σ_pΓ=∫_Dcurl(Γ)· n dσ=0 where n denotes the corresponding normal to D and where we used that Γ is curl-free. However, due to the boundary regularity, we only know that Γ∈⋂_1≤ p<∞W^1,p𝒱(Ω) so that the classical Stokes' theorem is not immediately applicable. To bypass this problem one can exploit the fact that Γ is in fact analytic within Ω, since it is a weak solution of ΔΓ=0 in Ω. We can then fix a (compactly supported) C^∞-vector field X defined on ℝ^3 which is everywhere inward pointing along Σ. If we let Ξ_τ denote the flow of X we obtain a C^1-curve σ_p(τ):=Ξ_τ∘σ_p⊂Ω which still bounds a C^1-disc within Ω since X is inward pointing. It then follows from an application of Stokes' theorem and interior regularity of Γ that ∫_σ_p(τ)Γ=0 for all τ>0. On the other hand, Sobolev embeddings tell us that Γ is continuous up to the boundary, so that it is not hard to see that ∫_σ_pΓ=lim_τ↘ 0∫_σ_p(τ)Γ=0 as previously suggested so that ∫_σ_pγ=0. Finally, if ∫_σ_tγ=0 would be true, it would follow, because σ_p and σ_t generate the first fundamental group, that ∫_σγ=0 for every closed loop σ so that a standard construction would imply that γ is a gradient field implying that γ=0 since γ is div-free and hence L^2-orthogonal to the gradient fields. Since γ≠ 0 we conclude that γ/∫_σ_tγ satisfies the defining equations of γ_t and so by uniqueness γ_t=γ/∫_σ_tγ. Using this, we obtain from (<ref>) and <Ref>
	
    α=Q(v)|Σ|∫_σ_tγ.

	Now, since γ and γ̃ form a basis of ℋ(Σ) we can also express γ_p=μγ+λγ̃ for suitable μ,λ∈ℝ. We find 1=∫_σ_pγ_p=λ∫_σ_pγ̃ so that λ=1/∫_σ_pγ̃ where ∫_σ_pγ̃≠ 0 because otherwise γ and γ̃ would be linearly dependent. Lastly, 0=∫_σ_tγ_p=μ∫_σ_tγ+∫_σ_tγ̃/∫_σ_pγ̃. We can solve this for μ and conclude from (<ref>) and <Ref>
	
    β=∫_Σγ_p/λ· v-μ/λγ· vdσ(x)=|Σ|(∫_σ_pγ̃)(P(v)+∫_σ_tγ̃/∫_σ_pγ̃Q(v))

	where we used that γ_t=γ/∫_σ_tγ. Inserting the expressions (<ref>) and (<ref>) into (<ref>) yields the result.
	
	Finally, if σ_t bounds a surface 𝒜 outside of Ω, the result follows from <Ref>.



 §.§ Proof of <Ref>


	We will divide the proof in several steps. We at first consider the divergence-free vector field w∈ C^0,1𝒱_0(Σ). We recall that by assumption there exists a positive function f∈ C^0,1(Σ,(0,∞)) such that v=w/f is rectifiable. Then in the last step we show that the rotational transform of v coincides with that of w which will conclude the proof.
	Step 1: We observe that v being rectifiable implies that either all of its field lines are closed or none of them are. Assume in the first step that the field lines of v are all closed. Then by the rectifiability property it follows further that all field lines are of the same type, i.e. if σ_x is a field line of v starting at x∈Σ and we write σ_x=P_xσ_p⊕ Q_xσ_t for suitable P_x,Q_x∈ℤ (and where σ_p,σ_t denote the fixed purely poloidal and toroidal curve respectively), then in fact P_x and Q_x are independent of the chosen point x. Since w and v have the same field lines, which are merely traced out with a different speed, we conclude that also all field lines of w are of the type Pσ_p⊕ Qσ_t with the same P,Q∈ℤ (which are independent of x and we thus drop the subscript x). We recall (<ref>) and see that if for a given x∈Σ we let τ_x>0 denote the period of the field line of w starting at x, then q̂(x)=Q/τ_x and with the same reasoning p̂(x)=P/τ_x where q̂,p̂ are the quantities associated with w. Further, by assumption, Q(w)≠ 0 which by definition implies 0≠∫_Σq̂(x)dσ(x)=Q∫_Σ1/τ_xdσ(x) and consequently Q≠ 0 so that w has a well-defined rotational transform at every x∈Σ and by definition ι_w(x)=p̂(x)/q̂(x)=P/Q since q̂(x)=Q/τ_x, p̂(x)=P/τ_x. Finally, since p̂(x)=P/Qq̂ and P/Q is a constant, we conclude P(w)=P/QQ(w) so that ι_w(x)=P(w)/Q(w) is independent of x and the statement follows from <Ref>.
	Step 2: Now we assume that the field lines of w are not closed. We recall that if we let μ denote the normalised surface measure on Σ, then an area preserving flow ψ_t is ergodic if for any (Borel-)measurable A⊂Σ with ψ_-t(A)⊂ A for all t≥ 0 we have μ(A)=0 or μ(A)=1.
	
	We claim first that the flow ψ_t of w is ergodic. Since w is div-free the flow is clearly area preserving. On the other hand, since v=w/f where f is a strictly positive function, it is easy to see that if Ψ_t denotes the flow of v, the condition ψ_-t(A)⊂ A for all t≥ 0 implies Ψ_-t(A)⊂ A for all t≥ 0. We have to prove that this implies that either A or Σ∖ A is a null-set. However, since diffeomorphisms preserve null-sets it is enough, due to the rectifiability of v, to see that the linear flow with irrational slope is ergodic on the flat torus which is a well-known fact, see <cit.>. Then the ergodicity and the ψ_t invariance of q̂ and p̂ respectively imply that q̂ and p̂ are both constant a.e.. Then by definition we find P(w)=p̂(x), Q(w)=q̂(x) for a.e. x and since by assumption Q(w)≠ 0 we see that w has a well-defined rotational transform given by ι_w(x)=p̂(x)/q̂(x)=P(w)/Q(w) so that the statement now follows once more from <Ref>.
	Step 3: Here we prove that ι_v(x)=ι_w(x) for a.e. x∈Σ, which will complete the proof.
	
	We recall the definition p̂_v(x)=lim_T→∞1/T∫_σ_x^v[0,T]γ_p, where the subscript indicates that p̂ refers to the poloidal twists of v and where σ_x^v denotes the field line of v starting at x. Further recall that w=fv for a strictly positive function f∈ C^0,1(Σ,(0,∞)). We can then consider 
	
    f̂(x):=lim_S→∞1/S∫_0^Sf(σ_x^w(t))dt,

where σ_x^w denotes the field line of w starting at x. By standard ergodic theory, c.f. <cit.>, this defines an element of L^1(Σ) since the flow of w is area-preserving. We observe that f is bounded above and away from zero so that 0≤ c_1≤f̂(x)≤ c_2<∞ for suitable constants 0<c_1≤ c_2 and all x∈Σ. Further, since v and w have the same field lines we may parametrise σ^v_x(t) via w so that we can write σ_x^v[0,T]=σ_x^w[0,S] for a suitable S>0 (in case that σ_x^v is periodic we pick S such that σ_x^w[0,S] passed x the same amount of times as σ_x[0,T] did). We therefore obtain
	
    p̂_v(x)=lim_T→∞1/T∫_σ_x^v[0,T]γ_p=lim_T→∞1/T∫_σ_x^w[0,S]γ_p

	where we note that S depends on T. To establish a relation between T and S we observe that T=∫_σ_x^v[0,T]v/|v|^2 and therefore we can write by choice of S
	
    T=∫_σ_x^w[0,S]v/|v|^2=∫_0^Sf(σ_x^w(t))dt

	where we used that w=fv. Since f is bounded away from zero and bounded above we see that T→∞⇔ S→∞. Inserting this relation into (<ref>) we find
	
    p̂_v(x)=lim_T→∞S/T·1/S∫_σ_x^w[0,S]γ_p=p̂_w(x)/f̂(x).

	In the same way one proves that q̂_v(x)=q̂_w(x)/f̂(x) so that in particular v has a well-defined rotational transform if and only if w has a well-defined rotational transform and we obtain ι_v(x)=p̂_v(x)/q̂_v(x)=p̂_w(x)/q̂_w(x)=ι_w(x) as claimed.



 §.§ Proof of <Ref>


	Case (i): If Σ≅ S^2 it follows from <Ref> that ℋ(w)=0 for all w∈ L^2𝒱_0(Σ) which implies the claim.
	Case (ii): Given any w∈ L^2𝒱_0(Σ) we can express w by means of the Hodge decomposition theorem as w=grad_Σ(f)×𝒩+γ for a suitable f∈ H^1(Σ) and γ∈ℋ(Σ). It then follows from <Ref> that if ℋ(w)≠ 0, we must have γ≠ 0. In addition, if ℋ(w)>0, the L^2(Σ)-orthogonality of the Hodge-decomposition implies that if grad_Σ(f)≠ 0 we must have ℋ(w)/w^2_L^2(Σ)<ℋ(w)/γ^2_L^2(Σ)=ℋ(γ)/γ^2_L^2(Σ) and that therefore w could not have been a maximiser. We will now at first show that there exists some w∈ L^2𝒱_0(Σ) with ℋ(w)>0 which by the previous argument will allow us to look for a maximiser within the space ℋ(Σ). To see this we let Ω denote the bounded domain bounded by Σ and fix any element Γ∈ℋ_N(Ω)∖{0} (the space of curl- and div-free H^1(Ω) vector fields which are tangent to Σ). Further, we let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ) and we define γ̃:=γ×𝒩. We recall that we have shown in the proof of <Ref> that 2ℋ_c(γ,γ̃)=γ^2_L^2(Σ) (ℋ_c being the cross-helicity) and that <Ref> states that ℋ(γ̃)=0. We can now let β≠ 0 be any constant and set α:=1/|β|. We define γ̂:=αγ+βγ̃ and observe that the previous considerations yield
	
    ℋ(γ̂)=α^2ℋ(γ)+2αβℋ_c(γ,γ̃)+β^2ℋ(γ̃)=α^2ℋ(γ)+αβγ^2_L^2(Σ)=1/β^2ℋ(γ)+sign(β)γ^2_L^2(Σ)

	where sign(β) equals +1 if β>0 and equals -1 if β<0. Letting β≫1 or β≪ -1 we see that there is some γ̂_± with ℋ(γ̂_±)· (±1)>0. Then we are left with observing that the functional {γ̂∈ℋ(Σ)|γ̂_L^2(Σ)=1}→ℝ,γ̂↦ℋ(γ̂) admits a (necessarily positive) global maximum since the unit sphere in finite dimensional vector spaces is compact. The claim follows then from the scaling properties of the helicity, i.e. ℋ(λ v)=λ^2ℋ(v) for all λ∈ℝ and v∈ L^2𝒱_0(Σ).



 §.§ Proof of <Ref>


	We recall that we let π:L^2𝒱(Σ)→ L^2𝒱_0(Σ) denote the L^2(Σ)-orthogonal projection from the space of square integrable vector fields on Σ into the space of square-integrable div-free fields on Σ. We prove first that π∘BS_Σ:L^2𝒱_0(Σ)→ L^2𝒱_0(Σ) is self-adjoint with respect to L^2(Σ). This is easy to see because for every v,w∈ L^2𝒱_0(Σ) we have ⟨ v,(π∘BS_Σ)(w)⟩_L^2(Σ)=⟨ v,BS_Σ(w)⟩_L^2(Σ) and one can then write out the definitions, use the cyclic property of the Euclidean inner product and Fubini's theorem to conclude the self-adjointness.
	
	We will now show that the image of π∘BS_Σ is finite dimensional and therefore this operator must be compact. Indeed, it follows from <Ref> that for any f∈ H^1(Σ) and w∈ L^2𝒱_0(Σ), we have ⟨grad_Σ(f)×𝒩,(π∘BS_Σ(w))⟩_L^2(Σ)=⟨grad_Σ(f)×𝒩,BS_Σ(w)⟩_L^2(Σ)=ℋ_c(grad_Σ(f)×𝒩,w)=0 and so the Hodge decomposition theorem implies that (π∘BS_Σ)(w)∈ℋ(Σ) for every w∈ L^2𝒱_0(Σ) which is a finite dimensional space. It now follows from the spectral theorem for compact, self-adjoint operators that π∘BS_Σ admits a discrete spectrum which accumulates at most at zero and a corresponding eigenbasis which together with Ker(π∘BS_Σ) spans the space L^2𝒱_0(Σ). According to the proof of <Ref> we see that there exist elements v,w∈ L^2𝒱_0(Σ) with ℋ(v)>0 and ℋ(w)<0. Since ℋ(v)=⟨ v,(π∘BS_Σ)(v)⟩_L^2(Σ) and similarly for w, it must be the case that π∘BS_Σ admits a positive and negative eigenvalue and since the spectrum accumulates at most at zero there must exist a largest positive and smallest negative eigenvalue. It then follows from the eigenspace decomposition that v∈ L^2𝒱_0(Σ) with v_L^2(Σ)=1 maximises helicity among all other such fields if and only if v is an eigenfield of π∘BS_Σ corresponding to the largest positive eigenvalue. Due to the scaling properties of helicity the same remains true for the maximisation of the Rayleigh quotient in (<ref>).
	
	

 §.§ Proof of <Ref>

	Before we come to the proof of <Ref> we prove the claim in <Ref>.
	
		
		Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then
		
    inf_T^2≅Σ∈ C^1,1Λ(Σ)=inf_T^2≅Σ∈ C^1,1,|Σ|=1Λ(Σ) and sup_T^2≅Σ∈ C^1,1Λ(Σ)=sup_T^2≅Σ∈ C^1,1,|Σ|=1Λ(Σ).

	
	
	We first prove that for given Σ and λ>0 we have Λ(Σ)=Λ(Σ_λ) where Σ_λ:={λ x| x∈Σ}. To see this we can start with an arbitrary vector field w∈ L^2𝒱_0(Σ) and define w_λ(x):=w(x/λ) for x∈Σ_λ. One easily verifies that w_λ∈ L^2𝒱_0(Σ_λ) and that accordingly the map w↦ w_λ gives rise to an isomorphism between L^2𝒱_0(Σ) and L^2𝒱_0(Σ_λ) (note that Σ=(Σ_λ)_1/λ). Then, using the change of variable formula and noting that ψ_λ:Σ→Σ_λ, x↦λ x provides an (orientation-preserving) diffeomorphism between Σ and Σ_λ, we find w_λ^2_L^2(Σ_λ)=λ^2w^2_L^2(Σ) for all w∈ L^2𝒱_0(Σ). As for helicity, we can write, keeping in mind that we perform a double integration,
	
    ℋ(w_λ)=1/4π∫_Σ_λ∫_Σ_λw_λ(x)·(w_λ(y)×x-y/|x-y|^3)dσ(y)dσ(x)
    		
    
    		=λ^4/4π∫_Σ∫_Σw(q)·(w(p)×λ q-λ p/|λ q-λ p|^3)dσ(p)dσ(q)=λ^2ℋ(w).

	Since w↦ w_λ defines an isomorphism we obtain
	
    Λ_±(Σ_λ)=max_w̃≠ 0
    w̃∈ L^2𝒱_0(Σ_λ)±ℋ(w̃)/w̃^2_L^2(Σ_λ)=max_w≠ 0
     w∈ L^2𝒱_0(Σ)±ℋ(w_λ)/w_λ^2_L^2(Σ_λ)=max_w≠ 0
     w∈ L^2𝒱_0(Σ)±ℋ(w)/w^2_L^2(Σ)=Λ_±(Σ)

	and therefore Λ(Σ_λ)=Λ(Σ) for all λ>0. We can now take sequences (Σ_n)_n approaching either the infimum or the supremum on the left hand side of the statement of the lemma respectively and rescale the Σ_n by appropriate λ_n>0 to normalise their area which in combination with Λ(Σ_λ)=Λ(Σ) immediately implies the claim.
	
	
		We recall that according to <Ref> and <Ref> the L^2(Σ)-normalised vector fields γ_+,γ_- realising the Rayleigh quotient (<ref>) are elements of ℋ(Σ) and eigenfields of π∘BS_Σ where π:L^2𝒱(Σ)→ L^2𝒱_0(Σ) denotes the L^2(Σ)-orthogonal projection onto L^2𝒱_0(Σ). In addition, while proving the compactness of π∘BS_Σ during the course of the proof of <Ref>, we also showed that π∘BS_Σ maps L^2𝒱_0(Σ) onto ℋ(Σ). Therefore, the operator π∘BS_Σ:ℋ(Σ)→ℋ(Σ) is well-defined and the Rayleigh quotient (<ref>) is realised by some eigenfields of this restricted operator. However, the Hodge isomorphism <cit.>, implies (ℋ(Σ))=(H^1_dR(Σ))=2, where H^1_dR(Σ) denotes the first de Rham cohomology group and where we used that Σ≅ T^2. It follows further from <Ref> that π∘BS_Σ admits at least one positive and one negative eigenvalue. Consequently, π∘BS_Σ|_ℋ(Σ) is a self-adjoint operator on the 2-dimensional Hilbert space (ℋ(Σ),⟨·,·⟩_L^2(Σ)). It therefore admits precisely two eigenvalues and thus one eigenvalue is strictly positive λ_+(Σ)>0 and one eigenvalue is negative λ_-(Σ)<0 with corresponding (L^2(Σ)-orthonormal) eigenfields γ_+(Σ) and γ_-(Σ) respectively. It follows immediately from the eigenspace decomposition that Λ_+(Σ)=ℋ(γ_+)=λ_+(Σ) and Λ_-(Σ)=-ℋ(γ_-)=-λ_-(Σ)=|λ_-(Σ)|. In order to obtain a relation between λ_+(Σ) and λ_-(Σ) we will now find an explicit matrix representation of (π∘BS_Σ)|_ℋ(Σ). To this end we let Ω⊂ℝ^3 denote the bounded domain which is bounded by Σ, ∂Ω=Σ. Then we can fix any Γ∈ℋ_N(Ω)∖{0} and as usual let γ∈ℋ(Σ) denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ). Upon rescaling we may assume that γ is L^2(Σ)-normalised. We then define γ̃:=γ×𝒩 where 𝒩 is the outward unit normal and note that γ̃∈ℋ(Σ) is also L^2(Σ)-normalised. We then observe that
		
    (π∘BS_Σ)(γ)=⟨ (π∘BS_Σ)(γ),γ⟩_L^2(Σ)γ+⟨ (π∘BS_Σ)(γ),γ̃⟩_L^2(Σ)γ̃=ℋ(γ)γ+ℋ_c(γ,γ̃)γ̃.

		Similarly we find (π∘BS_Σ)(γ̃)=ℋ_c(γ̃,γ)γ+ℋ(γ̃)γ̃ and therefore π∘BS_Σ has the following matrix representation with respect to the basis ℬ:={γ,γ̃}
		
    ℳ:=(π∘BS_Σ)_ℬ^ℬ=[      ℋ(γ) ℋ_c(γ,γ̃); ℋ_c(γ̃,γ)     ℋ(γ̃) ].

		It now follows from <Ref> that ℋ(γ̃)=0 and it follows from the first step in the proof of <Ref> and the symmetry of the Biot-Savart operator that 2ℋ_c(γ̃,γ)=2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)=1 due to the normalisation of γ. We obtain ℳ=[ ℋ(γ)  1/2;  1/2    0 ] and conclude 
		
    λ_+(Σ)λ_-(Σ)=(ℳ)=-1/4⇔Λ_+(Σ)Λ_-(Σ)=1/4.

		Now if both Λ_+(Σ) and Λ_-(Σ) would be strictly smaller than 1/2 it would contradict (<ref>) and therefore we must have Λ(Σ)≥1/2.
		
		Lastly we observe that Λ_+(Σ)-Λ_-(Σ)=λ_+(Σ)+λ_-(Σ)=Trace(ℳ)=ℋ(γ). So if ℋ(γ)=0 we see that Λ_+(Σ)=Λ_-(Σ) and then (<ref>) becomes Λ^2(Σ)=1/4⇒Λ(Σ)=1/2. Conversely, if Λ(Σ)=1/2 we note that (<ref>) can be equivalently expressed as Λ(Σ)min{Λ_+(Σ),Λ_-(Σ)}=1/4 and that therefore min{Λ_+(Σ),Λ_-(Σ)}=1/2=Λ(Σ) and so Λ_+(Σ)=Λ_-(Σ) so that 0=Trace(ℳ)=ℋ(γ).
	
	

 §.§ Proof of <Ref>

	
		According to <Ref> we only need to prove that ℋ(γ)=0 whenever Σ is a C^1,1-rotationally symmetric torus where we recall that γ is the L^2(Σ)-orthogonal projection of any element Γ∈ℋ_N(Ω)∖{0} onto ℋ(Σ) and where Ω denotes the bounded domain with boundary Σ. Upon applying isometries to Σ we may assume that the axis of symmetry of Σ is the z-axis. One can then consider the vector field Y(x,y,z):=(-y,x,0)^tr defined on all of ℝ^3 and which induces isometries. Since Σ is a rotationally symmetric regular torus, it does not intersect the axis of symmetry so that in particular Ω does not intersect it either. One can therefore consider the vector field Γ:=Y/|Y|^2 on Ω which can be easily verified to satisfy curl(Γ)=0 and div(Γ)=0. In addition, since Y generates rotations around the z-axis, we find Y∥Σ and hence Γ∥Σ so that Γ∈ℋ_N(Ω)∖{0}. Since Γ is curl-free and tangent to Σ it is standard that Γ|_Σ is also curl-free as a vector field on Σ. Finally we note that Y induces isometries so that it is a Killing field. But the restriction of Killing fields to invariant surfaces remain Killing so that one finds that Y|_Σ is div-free. In addition, a direct calculation yields Y·grad(|Y|^2)=0 on all of ℝ^3 so that the product rule gives us div_Σ(Γ|_Σ)=0. Consequently Γ|_Σ∈ℋ(Σ) and so γ=Γ|_Σ. Lastly we note that Γ|_Σ and Y|_Σ have the same field lines which are all mutually unlinked and periodic with a uniformly bounded period. It then follows from the linking interpretation of helicity, <Ref> and <Ref>, that ℋ(γ)=0 so that the claim follows from <Ref>.
	
	

 §.§ Proof of <Ref>

	
		We recall that we call an element j∈ L^2𝒱_0(Σ) simple, if it satisfies ∫_Σj·γ_tdσ=0 or equivalently, c.f. <Ref>, Q(j)=0 where γ_t is the unique element of ℋ(Σ) satisfying ∫_σ_pγ_t=0 and ∫_σ_tγ_t=1 (recall also that the notion of being simple is independent of the chosen curves σ_p,σ_t). We have to show that for any given j∈ L^2𝒱_0(Σ) there exists some j^'∈ L^2𝒱_0(Σ) which is simple and satisfies BS(j)(x)=BS(j^')(x) for all x∈Ω, where BS(j)(x):=1/4π∫_Σj(y)×x-y/|x-y|^3dσ(y) corresponds physically to the magnetic field inside the bounded domain Ω bounded by Σ which is induced by the current density j∈ L^2𝒱_0(Σ). So let j∈ L^2𝒱_0(Σ) be given, it then follows from <cit.> that there exists some j_0∈Ker(BS)∖{0}⊂ L^2𝒱_0(Σ)∩⋂_0<α<1C^0,α𝒱(Σ) so that BS(j+α j_0)=BS(j) for every α∈ℝ. Further, it was shown in the first step of the proof of <cit.> that if we fix any Γ∈ℋ_N(Ω)∖{0}, then j_0 can be chosen such that ∫_Σj_0(y)·Γ(y)dσ(y)=Γ^2_L^2(Ω)≠ 0. Now we observe first that curl(Γ)=0 in Ω implies that Γ|_Σ is curl-free on Σ and hence we can use the Hodge decomposition theorem to express Γ=grad_Σ(f)+γ for suitable f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ). Since j_0 is div-free it is L^2(Σ)-orthogonal to all gradient fields, which yields ∫_Σj_0·γ dσ=∫_Σj_0·Γ dσ≠ 0. Finally we recall that we have shown in the proof of <Ref> that γ_t=γ/∫_σ_tγ and consequently ∫_Σj_0·γ_tdσ≠ 0. Hence, by linearity, we may always find a suitable α_j∈ℝ such that ∫_Σ(j+α_j j_0)·γ_tdσ=0 so that we may set j^':=j+α_jj_0 and the regularity claims follow immediately from the regularity of j_0. Finally, we need to show that such a j^' is unique. To this end, suppose j^',ĵ∈ L^2𝒱_0(Σ) are simple and satisfy BS(j^')=BS(ĵ). Then j̅:=j^'-ĵ is also simple and contained in the kernel of BS. Therefore j̅∈Ker(BS) and consequently there exists some β∈ℝ with j̅=β j_0 where j_0 is chosen like above. In particular, ∫_Σj_0·γ_tdσ≠ 0 so that the simplicity of j̅ implies β=0 and consequently j^'=ĵ.
	
	

 §.§ Proof of <Ref>

	
		It follows from <cit.> that for every B_T∈ L^2ℋ(P) and every ϵ>0 there exists some j∈ L^2𝒱_0(Σ) with BS(j)-B_T_L^2(P)≤ϵ. Then <Ref> implies that there exists j^'∈ L^2𝒱_0(Σ) which is simple and with BS(j)=BS(j^') which concludes the proof.
	
	

 §.§ Proof of <Ref>

	
		The proof is in essence identical to the proof of <cit.>. For convenience of the reader we present it here nonetheless. Given B_T∈ L^2ℋ(P) we can fix any ϵ>0. According to <Ref> we can find some j_ϵ∈ L^2𝒱_0(Σ) which is simple and such that BS(j_ϵ)-B_T_L^2(P)^2≤ϵ. Then, by definition, j_λ^0 is the unique global minimiser of C_0(λ;B_T) (recall (<ref>)) so that
		
    C_0(λ;B_T)≤BS(j_ϵ)-B_T^2_L^2(P)+λj_ϵ^2_L^2(Σ)≤ϵ+λj_ϵ^2_L^2(Σ)

		by choice of j_ϵ. Further, we note that
		
    BS(j_λ^0)-B_T^2_L^2(P)≤BS(j_λ^0)-B_T^2_L^2(P)+λj^0_λ^2_L^2(Σ)=C_0(λ;B_T)

		and hence
		
    BS(j^0_λ)-B_T^2_L^2(P)≤ϵ+λj_ϵ^2_L^2(Σ).

		We recall that j_ϵ depends only on B_T and ϵ but is independent of λ. Therefore we may take the limit and obtain
		
    0≤lim inf_λ↘ 0BS(j^0_λ)-B_T^2_L^2(P)≤lim sup_λ↘ 0BS(j^0_λ)-B_T^2_L^2(P)≤ϵ.

		Since ϵ>0 was arbitrary we conclude lim_λ↘ 0BS(j^0_λ)-B_T_L^2(P)=0.
	


§ ACKNOWLEDGEMENTS

I would like to thank Ugo Boscain and Mario Sigalotti for fruitful discussions about the interpretation of surface helicity, in particular for suggesting to try to artificially close the field lines as depicted in <Ref>. This work has been supported by the Inria AEX StellaCage.


§ CONTINUITY OF SURFACE BIOT-SAVART OPERATOR



	
	<Ref> is well-defined, i.e. the Biot-Savart operator indeed defines a q-integrable vector field for any choice of orthonormal frame and it is independent of the chosen frame.


	First let us suppose that we have shown that
	
    ∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x)

	is finite for a.e. x in the domain of definition of any fixed orthonormal frame E_i. Then, if E_i and Ẽ_j are any two orthonormal frames whose domains of definition overlap, then on this overlap the integrals will exist almost surely for any x in the overlap. For any such fixed x we then have
	
    ∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x)
    		
    
    		=∑_j=1^2∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))(E_i(x)·Ẽ_̃j̃(x))Ẽ_j(x)
    		
    
    		=∑_j=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)·∑_i=1^2[(E_i(x)·Ẽ_̃j̃(x))E_i(x)])Ẽ_j(x)
    		
    
    		=∑_j=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)·Ẽ_j(x))Ẽ_j(x)

	and so the value of the Biot-Savart operator is independent of the chosen local orthonormal frames.
	
	In order to establish the q-integrability we define u(y):=𝒩(y)× v(y) and fix any w∈ T_xΣ. We observe that because v(y) is tangent to Σ we have v(y)=u(y)×𝒩(y). Then, using the vector triple product rule, we find
	
    ∫_Σ v(y)×x-y/|x-y|^3dσ(y)· w=∫_Σ(u(y)·x-y/|x-y|^3)𝒩(y)-(𝒩(y)·x-y/|x-y|^3)u(y)dσ(y)· w.

	It is well-known that there is some c>0 satisfying |x-y/|x-y|^3·𝒩(y)|≤c/|x-y| for all y,x∈Σ with y≠ x, c.f. <cit.>. On the other hand, we have, since w is orthogonal to 𝒩(x),
	
    |(u(y)·x-y/|x-y|^3)(𝒩(y)· w)|=|(u(y)·x-y/|x-y|^3)(𝒩(y)-𝒩(x))· w|≤ L|v(y)||w|/|x-y|

	where L is the Lipschitz character of 𝒩 and we used that |u(y)|=|v(y)|. Therefore, if |w|=1, then we see that there is some constant c_0>0 which is independent of v such that
	
    |∫_Σ v(y)×x-y/|x-y|^3dσ(y)· w|≤ c_0∫_Σ|v(y)|/|x-y|d σ(y).

	It follows easily from the dominated convergence theorem that if 1≤β <2 then the map Σ→ℝ, x↦∫_Σ1/|x-y|^βdσ(y) is continuous so that it is globally bounded on Σ. Hence, the above bound ensures that BS_Σ(v)∈ L^∞𝒱(Σ) for any v∈ L^p𝒱(Σ) with p>2 by an application of Hölder's inequality.
	
	If v∈ L^2𝒱(Σ), then in particular v∈ L^p𝒱(Σ) for any 1<p<2 and so the claim will follow from the case p<2 after observing that 2p/2-p converges to infinity as p↗ 2.
	
	Suppose now that 1<p<2. In that case one can use a partition of unity to localise the problem and hence reduce the situation to an application of the standard Hardy-Littlewood-Sobolev inequality, c.f. <cit.>.



§ PHYSICAL ROTATIONAL TRANSFORM


To simplify the discussion, we consider here only "unknotted" smooth solid tori Ω⊂ℝ^3 in the sense that we demand that there exists an orientation preserving C^∞-diffeomorphism Ψ:ℝ^3→ℝ^3 such that Ω=Φ(Ω_S) where Ω_S:={(x,y,z)^tr∈ℝ^3| (√(x^2+y^2)-2)^2+z^2≤1} is the standard rotationally symmetric solid torus bounded by a torus with minor radius r=1 and major radius R=2. We have the following parametrisation of Σ_S:=∂Ω_S, (ϕ,θ)↦μ(ϕ,θ)=((2+cos(θ))cos(ϕ),(2+cos(θ))sin(ϕ),sin(θ))^tr, 0≤ϕ,θ≤ 2π where ϕ is called the "toroidal" angle and θ is called the "poloidal angle". Further, we consider the following curves σ^S_p(τ):=(2+cos(τ),0,sin(τ))^tr, σ^S_t(τ):=(cos(τ),sin(τ),0)^tr, 0≤τ≤ 2π and call them the standard purely poloidal and standard purely toroidal loops. We then refer to σ_p:=Φ∘σ^S_p and σ_t:=Φ∘σ^S_t as a purely poloidal and a purely toroidal curve on Σ=∂Ω.

In addition, we observe that for any fixed ϕ∈ [0,2π), the curve σ_ϕ(τ):=μ(ϕ,τ), 0≤τ≤ 2π, is a closed circle (which bounds a disc within Ω_S). We call the family of images, Σ^ϕ_S:=σ_ϕ([0,2π]), (Σ^ϕ_S)_0≤ϕ<2π the standard foliation of Σ_S into poloidal sections. Accordingly we call (Σ_ϕ)_0≤ϕ<2π with Σ_ϕ:=Φ(Σ^S_ϕ) a foliation of Σ into poloidal sections.

Now one often assumes in plasma physics that the magnetic field lines of plasma equilibria B are transversal to the poloidal sections, i.e. never tangent to any such section, recall also <Ref> regarding the flow structure of plasma equilibria. Then, given a field line of B starting at some initial poloidal section Σ_ϕ_0, it will, due to the transversality assumption, continue to move along these poloidal sections Σ_ϕ until it returns to the initial poloidal section Σ_ϕ_0. During this toroidal turn, the field line will also have made a certain amount of poloidal turns (Δθ)_1. More precisely, we may write σ(τ)=Φ(σ_S(τ)) where σ_S(τ) is a curve on Σ_S parametrised by σ_S(τ)=μ(ϕ(τ),θ(τ)) for suitable smooth functions ϕ(τ) and θ(τ). The transversality condition demands that ϕ(τ) is a strictly increasing function in τ. Therefore there will be a unique 0<τ_1<∞ with ϕ(τ_1)=2π which corresponds to a full toroidal turn upon which we returned to Σ_ϕ_0. Then we set (Δθ)_1:=θ(τ_1). Similarly we can continue to follow our field line until we arrive again at Σ_ϕ_0 which uniquely determines a smallest time τ_1<τ_2<∞ with ϕ(τ_2)=4π. Accordingly we define (Δθ)_2:=θ(τ_2)-θ(τ_1) to be the amount of poloidal turns made during the second toroidal turn. Similarly we define (Δθ)_k to be the amount of poloidal turns during the k-th toroidal turn.

One way to define "physical rotational transform" of a magnetic field line σ starting at Σ_ϕ_0 is then as follows, c.f. <cit.>,

    ι_P(σ):=lim_n→∞∑_k=1^n(Δθ)_k/2π n

whenever the limit exists.

Suppose for now that σ is periodic, then it will be homotopic to Pσ_p⊕ Qσ_t for suitable P,Q∈ℤ. We observe that this means that σ performs precisely Q toroidal turns and P poloidal turns within one period. We are assuming (without loss of generality) that σ moves in the same toroidal direction as σ_t which corresponds to Q≥ 0 and the transversality assumption ensures Q≥ 1. We know that after every Q toroidal turns, we performed P poloidal turns so that ∑_i=1^kQ(Δθ)_i=2π kP. Thus, if kQ≤ n<(k+1)Q, we find ∑_i=1^n(Δθ)_i=2π kP+∑_i=kQ+1^n(Δθ)_i and we observe that -2π P≤∑_i=kQ+1^n(Δθ)_i≤ 2π P because we made less than Q additional toroidal turns and hence at most P poloidal turns. We note further that Q≤n/k≤ (1+1/k)Q so that n/k→ Q as n→∞. Hence, we overall arrive at

    ι_P(σ)=lim_n→∞∑_k=1^n(Δθ)_k/2π n=lim_n→∞(k/nP+R(n)/n)=P/Q

where R(n) is the remainder term which we have shown to be bounded in absolute value by 2π P. We recall that with our "mathematical" rotational transform ι we have, c.f. <Ref>, ι(σ)=∫_ΣB·γ_pdσ/∫_ΣB·γ_t dσ where B=fB and f>0 is a suitable positive function related to the pressure function of the plasma equilibrium and B is the underlying magnetic field. Most importantly, a field line σ of B starting at some point in Σ is of type Pσ_p⊕ Qσ_t if and only if the same is true for B. But then <Ref> and (<ref>) imply that ι(σ)=P/Q=ι_P(σ) so that the rotational transform agrees in this case, recall also <Ref>.

If the field line σ is not closed, we can close it artificially. The idea is that at time τ_n>0 when we return for the n-th time to the initial poloidal section Σ_ϕ_0 we close the curve by following the path σ_ϕ_0 from σ(τ_n) along the poloidal section until we reach the first point on Σ_ϕ_0 which was intersected by σ previously (recall that we will have visited Σ_ϕ_0 by that time precisely (n-1)-times resulting (due to the non-periodicity) in (n-1) intersection points and since Σ_ϕ_0 is a circle, we must necessarily arrive at such a point, provided n≥ 2). The important observation is that by means of this closing procedure we add at most an angle |α_n|≤ 2π to the total toroidal angle. If we denote by σ̃_n=σ[0,τ_n]⊕σ_ϕ_0 the corresponding concatenated path, we see that it will have made precisely n toroidal turns and P_n∈ℤ poloidal turns. Hence, if we let ι^n_P(σ):=∑_k=1^n(Δθ)_k/2π n then we obtain

    ι^n_P(σ)=P_n/n-α_n/2π n, |α_n|≤ 2π, ι_P(σ)=lim_n→∞ι^n_P(σ).

On the other hand, with the "mathematical" definition of rotational transform we find

    ι(σ)=lim_T→∞∫_σ[0,T]γ_p/∫_σ[0,T]γ_t

and know that this limit exists for a.e. x∈Σ at which the field line σ starts and that the limit is independent of the chosen sequence T→∞, c.f. <Ref>. We can therefore consider the sequence T=τ_n which diverges to +∞ as n→∞. But we observe that ∫_σ[0,τ_n]γ_p=∫_σ̃_nγ_p-∫_σ_ϕ_0[t^1_n,t^2_n]γ_p=P_n-r_n where we define the remainder r_n:=∫_σ_ϕ_0[t^1_n,t^2_n]γ_p and where we used that σ̃_n is closed and makes P_n poloidal turns. We observe further that |r_n| can be uniformly bounded because γ_p is globally bounded on Σ, σ̇_ϕ is globally bounded along any poloidal section and because by construction t^2_n-t^1_n=|α_n|≤ 2π is the additional toroidal angle which we need to sweep out to close our curve. Identically we can argue that ∫_σ[0,τ_n]γ_t=n+R_n where R_n is a uniformly (in n) bounded error term and we made use of the fact that σ̃_n makes n toroidal turns. We therefore arrive at

    ∫_σ[0,τ_n]γ_p/∫_σ[0,τ_n]γ_t=P_n-r_n/n+R_n=P_n/n-r_n/n/1+R_n/n=ι^n_P(σ)+α_n/2π n-r_n/2π/1+R_n/n.

Since the limit on the left hand side exists we conclude that also the physical rotational transform is well-defined and that ι(σ)=lim_n→∞∫_σ[0,τ_n]γ_p/∫_σ[0,τ_n]γ_t=lim_n→∞ι^n_P(σ)=ι_P(σ) and therefore the physical rotational transform and mathematical rotational transform always coincide and hence <Ref> provides a way to compute the rotational transform of a plasma equilibrium B, see also <Ref>.


§ EIGENFIELDS OF THE SURFACE BIOT-SAVART OPERATOR



	
	Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-torus and let Ω⊂ℝ^3 be the unique bounded domain whose boundary is Σ. Let further Γ∈ℋ_N(Ω)∖{0} and let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ) and set γ̃:=γ×𝒩 where 𝒩 denotes the outward unit normal. Further we scale Γ such that γ is L^2(Σ)-normalised. Then the set of vector fields v_±∈ℋ(Σ) realising the Rayleigh quotient Λ_±(Σ), c.f. (<ref>), are 1-dimensional subspaces E_±(Σ) of ℋ(Σ) respectively and spanned by the following vector fields
	
    E_+(Σ)=span{(√(ℋ^2(γ)+1)+ℋ(γ))γ+γ̃}, E_-(Σ)=span{γ-(√(ℋ^2(γ)+1)+ℋ(γ))γ̃}

	and we have
	
    Λ_+(Σ)=√(ℋ^2(γ)+1)+ℋ(γ)/2, Λ_-(Σ)=√(ℋ^2(γ)+1)-ℋ(γ)/2.



	We recall that we have shown in the proof of <Ref> that we may restrict attention to the operator (π∘BS_Σ)|_ℋ(Σ) which maps into ℋ(Σ) and has the following matrix representation with respect to the basis ℬ:={γ,γ̃}
	
    ℳ:=(π∘BS_Σ)^ℬ_ℬ=[ ℋ(γ)  1/2;  1/2    0 ].

	Computing the eigenvalues and eigenvectors of this matrix is a standard exercise and yields the result upon recalling that Λ_+(Σ)=λ_+(Σ) and Λ_-(Σ)=-λ_-(Σ) where λ_+(Σ),λ_-(Σ) denote the positive and negative eigenvalue of ℳ respectively and that the eigenvectors of ℳ precisely coincide with the vector fields v_± of interest.


	We note that we have already argued in the proof of <Ref> that ℋ(γ)=0 whenever Σ is a rotationally symmetric C^1,1-torus. In addition, upon applying isometries and therefore assuming that the z-axis is the symmetry axis, we had found that γ=Γ|_Σ where Γ=Y/|Y|^2 and Y(x,y,z)=(-y,x,0)^tr it the vector field inducing rotations around the z-axis. Hence <Ref> tells us that in this case Λ_±(Σ)=1/2 and that E_±(Σ)={γ±γ̃} where γ is explicitly known and γ̃=γ×𝒩 is determined by the outward unit normal field of Σ which can for instance be obtained from a globally defining function of Σ.

The downside of <Ref> is that we still need to compute ℋ(γ) which is again a non-trivial task. The following proposition reduces the problem to compute certain (simple) line and surface integrals.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3 and suppose that σ_t is a toroidal curve which bounds a bounded C^1,1-surface 𝒜 outside of Ω, i.e. 𝒜⊂ℝ^3∖Ω and ∂𝒜=σ_t. Let Γ∈ℋ_N(Ω)∖{0} be fixed and let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ). Further let γ̃:=γ×𝒩. Then we have
	
    ℋ(γ)=-∫_σ_tγ̃/∫_σ_tγγ^2_L^2(Σ).



	We follow the ideas of the proof of <Ref>. We observe that we want to compute 4πℋ(γ)=4πℋ_c(γ,γ). We recall that γ̃=γ×𝒩 and consequently γ=𝒩×γ̃. We can then follow the reasoning of Step 1 in the proof of <Ref> where we computed ℋ_c(γ̃,γ). We observe that since γ=-(γ̃×𝒩) we will get an additional minus sign in the final expression. Apart from that only γ(x) has to be replaced by γ̃(x) throughout the computations which eventually leads us to the following identity
	
    4πℋ(γ)=-∫_Σγ(y)·[4πγ̃(y)/2+∫_Ωdiv(ṽ)(x)x-y/|x-y|^3d^3x+∫_Ωcurl(ṽ)(x)×x-y/|x-y|^3d^3x]dσ(y)
    		
    
    		=-∫_Σγ(y)·[∫_Ωdiv(ṽ)(x)x-y/|x-y|^3d^3x+∫_Ωcurl(ṽ)(x)×x-y/|x-y|^3d^3x]dσ(y)

	where we used that γ and γ̃ are perpendicular and where ṽ is any H^1-vector field on Ω such that ṽ^∥=γ̃. We note that if 𝒩×ṽ=γ, then due to the relation of γ and γ̃ this implies ṽ^∥=γ̃. It then follows from <cit.> and its proof and the regularity of γ that there exists some ṽ∈ W^1,2𝒱(Ω) satisfying 𝒩×ṽ=γ, div(ṽ)=0 in Ω and curl(ṽ)=Γ where Γ∈ℋ_N(Ω) satisfies Γ^2_L^2(Ω)=∫_Σγ·Γdσ. We note that in particular ∫_Σγ·Γdσ≥ 0 and since ∫_Σγ·Γ dσ=γ^2_L^2(Σ)>0 we must have, since ℋ_N(Ω) is spanned by Γ, Γ=Γ_L^2(Ω)/Γ_L^2(Ω)Γ from which we conclude Γ_L^2(Ω)=∫_Σγ·Γdσ/Γ_L^2(Ω)=∫_Σγ·Γ dσ/Γ_L^2(Ω). We hence arrive at curl(ṽ)=∫_Σγ·Γ dσ/Γ^2_L^2(Ω)Γ=γ^2_L^2(Σ)/Γ^2_L^2(Ω)Γ where we used that γ is the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ).
	
	Combining these considerations, we arrive at
	
	
	ℋ(γ)=γ^2_L^2(Σ)/Γ^2_L^2(Ω)∫_Σγ(y)·BS_Ω(Γ)(y)dσ(y)

	where BS_Ω(Γ)(y)=1/4π∫_ΩΓ(x)×y-x/|y-x|^3d^3x denotes the volume Biot-Savart operator.
	
	Since Γ is div-free and tangent to the boundary of Ω it is standard, <cit.>, that curl(BS_Ω(Γ))=Γ in Ω and curl(BS_Ω(Γ))=0 outside of Ω. We further observe that since curl(BS_Ω(Γ))=Γ∥Σ, the tangent part of the restriction of BS_Ω(Γ) to Σ is curl-free. In particular, according to the Hodge decomposition theorem it follows that the L^2(Σ)-orthogonal projection of (BS_Ω(Γ))^∥ onto ℋ(Σ) differs from the original tangential field (BS_Ω(Γ))^∥ only by a gradient field. Consequently, if we let σ_t denote the toroidal curve bounding a surface 𝒜 outside of Ω, we obtain, by virtue of Stokes' theorem,
	
		
		∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=∫_σ_tBS_Ω(Γ)=∫_𝒜curl(BS_Ω(Γ))·n dσ=0,
	
	where π_ℋ(Σ) denote the projection onto ℋ(Σ), where we used that σ_t is tangent to Σ, where n denotes the corresponding unit normal and where we used that curl(BS_Ω(Γ))=0 on ℝ^3∖Ω.
	
	Due to the scaling properties of helicity we may now without loss of generality assume that γ (and therefore γ̃) are L^2(Σ)-normalised and hence form an L^2(Σ)-orthonormal basis of ℋ(Σ). We can then express
	
		
		π_ℋ(Σ)((BS_Ω(Γ))^∥)=∫_Σγ·BS_Ω(Γ)dσγ+∫_Σγ̃·BS_Ω(γ)dσγ̃
	
	where we used that γ and γ̃ are tangent to Σ. Combining (<ref>),(<ref>) and (<ref>) (keeping in mind that we assume γ to be normalised) we arrive at
	
		
		0=∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Γ^2_L^2(Ω)ℋ(γ)∫_σ_tγ+∫_Σγ̃·BS_Ω(Γ)dσ∫_σ_tγ̃.
	
	Finally we recall that (BS_Ω(Γ))^∥ is curl-free in the weak sense on Σ and that so is Γ|_Σ because curl(BS_Ω(Γ))=Γ and curl(Γ)=0 in Ω are both tangent to Σ. Accordingly Γ×𝒩 and γ×𝒩=γ̃ differ only by a co-exact vector field and keeping in mind that (BS_Ω(Γ))^∥ is L^2(Σ)-orthogonal to the co-exact fields (as it is weakly curl-free on Σ) we may calculate
	
		∫_Σγ̃·BS_Ω(Γ)dσ=∫_Σγ̃·(BS_Ω(Γ))^∥dσ=∫_Σ(Γ×𝒩)·(BS_Ω(Γ))^∥dσ=∫_Σ(Γ×𝒩)·BS_Ω(Γ)dσ
		
		
		=∫_Σ(BS_Ω(Γ)×Γ)·𝒩dσ=∫_Ωdiv(BS_Ω(Γ)×Γ)d^3x=Γ^2_L^2(Ω),
	
	where we used the cyclic properties of the Euclidean product, that γ̃ it tangent to Σ and the standard calculus formula div(X× Y)=curl(X)· Y-X·curl(Y) for arbitrary C^1-vector fields X,Y on Ω. We can insert this identity in (<ref>) and finally arrive at
	
    0=Γ^2_L^2(Ω)(ℋ(γ)∫_σ_tγ+∫_σ_tγ̃)

	which immediately implies the claimed identity.

The following final result allows us to express the helicity of a vector field through Γ alone.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Suppose that σ_p is a purely poloidal curve which bounds a C^1,1-disc D⊂Ω and suppose that σ_t is a toroidal curve bounding a C^1,1-surface 𝒜⊂Ω^c. Let further Γ∈ℋ_N(Ω)∖{0} and let γ denote the projection of Γ|_Σ onto ℋ(Σ) and set γ̃:=γ×𝒩. Then
	
		
		∫_σ_tγ=∫_σ_tΓ, ∫_σ_pγ̃=Flux(Γ)/Γ^2_L^2(Ω)γ^2_L^2(Σ)
	
	where Flux(Γ):=∫_DΓ· n dσ and n is the unit normal on the disc D compatible with a chosen orientation of σ_p with regards to Stokes' theorem. In particular, if v∈ C^0,1𝒱_0(Σ), then
	
		
		ℋ(v)/|Σ|^2=Q(v)P(v)(∫_σ_tΓ)Flux(Γ)/Γ^2_L^2(Ω)=∫_Σv·Γdσ∫_Σv·BS_Ω(Γ)dσ/Γ^2_L^2(Ω)|Σ|^2.
	


	The first identity in (<ref>) follows from the fact that Γ|_Σ is curl-free in the weak sense and therefore Γ|_Σ and γ differ only by a gradient field, so that the fact that σ_t is a closed curve yields the result. We assume now without loss of generality that γ is L^2(Σ)-normalised. Then for the second identity in (<ref>) we may utilise (<ref>),(<ref>) and the fact that ∫_σ_pγ=0 to obtain
	
    Flux(Γ)=∫_DΓ· ndσ=∫_Dcurl(BS_Ω(Γ))· ndσ=∫_σ_pBS_Ω(Γ)=∫_σ_pπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Γ^2_L^2(Ω)∫_σ_pγ̃

	where we used that D⊂Ω, ∂ D=σ_p, that σ_p is tangent to Σ and that (BS_Ω(Γ))^∥ is curl-free in the weak sense.
	
	The first equality in (<ref>) is a direct consequence of (<ref>) and <Ref>. As for the second identity we recall that we have argued on several occasions that (∫_σ_tγ) γ_t=γ and we have seen that ∫_σ_pπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Flux(Γ) and according to (<ref>) ∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=0 so that by the defining properties of γ_p we find Flux(Γ)γ_p=π_ℋ(Σ)((BS_Ω(Γ))^∥). We can then utilise <Ref> and use the fact that v is tangent to Σ and div-free and that Γ and γ as well as (BS_Ω(Γ))^∥ and π_ℋ(Σ)((BS_Ω(Γ))^∥) only differ by a gradient field respectively which yields the last equality.

plain



 Paper Title: Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces.



Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces]Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces










vaughngzenda@cmail.carleton.ca

Robin Chhabra*robin.chhabra@carleton.ca

Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Dr, Ottawa, K1S 5B6, Ontario, Canada






   In this paper, we study an affine connection approach to realizing nonholonomic mechanical systems mediated by viscous friction forces with large coefficients, viewed as a singular perturbation of the nonholonomic system. We show that the associated slow manifold is represented coordinate-free as the image of a section over the nonholonomic distribution. We propose a novel invariance condition based on covariant derivatives and prove that this condition is equivalent to the classical invariance condition based on time derivatives. Accordingly, we propose a novel recursive procedure to approximate the slow manifold based on the covariant derivatives of a formal power series expansion of the section. Using this recurrence relation, we derive, up to second order, approximations of the slip velocities residing in the slow manifold, as well as the associated approximated dynamics up to first order. Lastly, we illustrate our approach with a case study of a vertical rolling disk. 








[
    *
    August 5, 2024
==================

Submitted to the Journal of Nonlinear Dynamics




§ INTRODUCTION

Rolling without slipping is a constraint that appears in many mechanical systems, such as vehicle systems, and is commonly viewed as an example of an ideal nonholonomic constraint <cit.>. Nonholonomic constraints normally refer to point-wise linear constraints on the velocity space at each configuration, which cannot be described by a derivative of a set of constraints on the configuration space <cit.>. Hence, such constraints are then viewed as a nonintegrable linear distribution of the tangent bundle to the configuration manifold <cit.>. However, the nonholonomic constraint forces are only valid for velocities in the distribution. In practice, nonholonomic constraints in vehicle systems are often violated, resulting in an accumulation of errors for standard nonholonomic model-based control laws. To model the velocities violating the nonholonomic constraints, the ideal nonholonomic constraint forces may be modelled as dissipative friction, whose limit at large coefficients approximates the ideal nonholonomic system <cit.>. The large friction coefficients introduce a steep linear relationship between the friction force and the motion of the contact point that is analogous to the linear regime of Bakker-Pacejka magic formula <cit.>. Mathematically, the system with large coefficients may be treated as a slow/fast singular perturbation where the dissipative friction forces act on a much faster time scale than the nominal variables resulting in an exponentially attractive invariant manifold. This manifold – often called the slow manifold – physically represents small violations of the nonholonomic constraints. 

Nonholonomic mechanical systems form a class of constrained systems whose velocity constraints have a natural differential geometric structure  <cit.>. Classically, the equations of motion are derived using Lagrange-d'Alembert's (L-d'A) variational principle – which is an adaptation of Hamilton's variational principle –  whereby the variations are constrained to the nonholonomic distribution and enforced with Lagrange multipliers that do no work on the distribution <cit.>. Some examples of nonholonomic constraints include rolling without slipping, no sliding, and conserved angular momenta, among
others <cit.>. The kinetic energy of the mechanical system may be interpreted geometrically as a Riemannian metric tensor <cit.>. Equivalent to the Lagrange-d'Alembert principle, Lewis and Bullo <cit.> use Riemannian geometry to model the nonholonomic dynamics with affine connections. An affine connection is a geometric object that enables a notion of differentiation of vector fields on a smooth manifold. A unique affine connection is derived from the metric tensor called the Levi-Civita affine connection that encodes Coriolis, centrifugal, and centripetal accelerations. In <cit.>, the nonholonomic constraints may be viewed as the kernel of a projection map onto the complement of the nonholonomic distribution. In the affine connection framework, <cit.>, the nonholonomic constraints are then enforced using Lagrange multipliers that are solved for using a covariant derivative of the complementary projection map. As such, Lewis introduces an affine connection called the nonholonomic connection, based on the Levi-Civita connection and the covariant of the orthogonal projection map, modelling the necessary accelerations required to maintain the nonholonomic constraints. The geometric interplay of affine connections and nonintegrable distributions is studied in detail in <cit.>. Other equivalent approaches to nonholonomic mechanics include symplectic geometry <cit.>, Poisson geometry <cit.>, Gauss's Principle of Least Constraint <cit.> and Kane's equations <cit.>.

Singularly perturbed systems (SPS) manifest in various domains of science and engineering, emerging when a dynamical system relies on scalable parasitic parameters like small time constants, masses, friction, . Accordingly, SPS have natural applications in electric circuits <cit.>, chemistry <cit.>, neuroscience <cit.>, among others. Slow/fast systems are a particular case of a SPS that involve two distinct time scales, delineated by a singular parameter ϵ. Tikhonov established that when the spectra of the linearized fast dynamics lies in the left half-plane, the flow converges to an invariant (slow) manifold as ϵ tends to 0 <cit.>. The functional relationship between the slow and fast variables is dictated by an invariance condition and a formal power series expansion is a common approach to approximate the slow manifold <cit.>. Fenichel examined the persistence of the invariant manifold under small perturbations of , utilizing the concept of Normally Hyperbolic Invariant Manifold (NHIM) and under the assumption of compactness in the state space <cit.>. In <cit.>, Fenichel utilizes a geometric approach to investigate asymptotic expansions of the graph representation of the invariant manifold. In the Riemannian geometric context, Eldering extends Fenichel's findings to non-compact state spaces, assuming regularity under bounded geometry conditions <cit.>. Eldering's work opens up a pathway for the coordinate-independent examination of such systems. If a mechanical system possesses a noncompact configuration space Q and is endowed with a symmetry group G, where the group action leads to a compact shape space Q/G <cit.>, then it follows that Q has bounded geometry <cit.>. In practical applications, numerous mechanical systems exhibit symmetry groups, as seen in mobile vehicles that yield compact shape spaces <cit.>. Hence, Eldering's geometric singular perturbation techniques <cit.> can be employed for the analysis of such systems.


The physical realization of nonholonomic mechanical systems by strong viscous friction forces was treated classically by Brendelev and Karapetian who mainly focused on convergence to the ideal nonholonomic dynamics <cit.>. More recently, Eldering used the geometric singular perturbation theory of NHIM's <cit.> to generalize these convergence results for systems with bounded geometry. Common to all these models, the contact forces are modelled as a Rayleigh dissipation function (RDF) defined by a symmetric (0,2)-tensor. Rayleigh dissipation functions
have been used to model surface contact forces following first physical principles <cit.>. To realize the nonholonomic constraints, the RDF is constructed such that its kernel is identified with the nonholonomic distribution ensuring that the contact forces do no work on the distribution, mimicking the L-d'A principle, yet the dynamics are exponentially attractive to the distribution. At the limit of large friction coefficients, Eldering proves the existence of a persistent exponentially attractive NHIM and shows that in a nonstandard frame spanning the distribution and its complement, the slow manifold may be represented as the graph over the nonholonomic variables <cit.>. It is shown that this graph may be represented as a formal power series, which is used to approximate the slow manifold up to first order <cit.> based on the classical invariance condition.

Antali and Stepan considered the physical realization of the rolling phenomenon of wheels by dry friction forces <cit.>. Their analysis is based on the crossing dynamics of extended Filippov systems <cit.>. A Filippov system is described by a piece-wise smooth vector field with discontinuities on a discontinuity manifold <cit.>. Extended Filippov (EF) systems are Filippov systems with codimension-2 discontinuity manifolds and describe Coulomb rigid body dynamics <cit.>. The EF approach has been used to analyze the spinning and sliding behavior of rigid bodies under contact <cit.>. Recently, bifurcation analysis of EF systems has also been developed to analyze slip and sliding dynamics of a wheel motion <cit.>. In contrast to the slow-fast realization by viscous friction forces <cit.>, the Coulomb friction in <cit.> is both nonsmooth and nonlinear in velocities. The wheel rolling phenomenon has also been captured as a limit of piece-wise continuous holonomic constraints <cit.>. Other historical approaches to the realization of nonholonomic constraints include coupling of an external force to generate nonholonomic dynamics <cit.>.



In this paper, we study realizations of nonholonomic systems as slow/fast systems using an affine connection approach inspired by the work of Bullo and Lewis <cit.>. In our geometric approach, we represent the slow manifold coordinate-independent as the image of a section over the nonholonomic distribution. We introduce a novel invariance based on a covariant derivative along the integral curves on the slow manifold. We prove that this novel invariance condition is equivalent to the invariance condition based on time derivatives <cit.>. We propose a recursion to approximate the velocities in the slow manifold based on covariant derivatives of geometric quantities. Using the recursion, we derive approximations of equations of motion. Lastly, we study an example system of a vertical rolling disk. The major contributions of this work are summarized as follows:

    
  * formulating a coordinate-free representation of the slow manifold as the image of a nonlinear section over the nonholonomic distribution, 
    
  * formulating a novel invariance condition based on covariant derivatives, and proving its equivalence to the classical invariance condition, 
    
  * proposing a novel recurrence relation to approximate the slip velocities described by the slow manifold,
    
  * using the recurrence relation to derive higher-order perturbations of the equations of motion.


The paper is organized as follows. Section <ref> provides a self-contained summary of the differential geometric structures used in the remainder of the article. This includes a review of Riemannian geometry, affine connection systems, nonholonomic affine connection systems, and the geometry of the second tangent bundle including the procedure to take covariant derivatives of general (nonlinear) tangent bundle maps. In Section <ref>, we derive the equations of motion for a slow fast realization of nonholonomic systems and represent the slow manifold as the image of a section. Using this model, in Section <ref>, we derive the relationship between the reaction forces and the slow manifold and calculate zeroth, first and second order approximations of the slip velocities, and zeroth and first-order dynamical equations of motion. Lastly, in Section <ref> we consider our approach with a case study of a vertical rolling disk. 








§ PRELIMINARIES
 
This section provides a self-contained review of mathematical notions used throughout the paper. For references on Riemannian geometry, geometric mechanics and related topics see <cit.> and references therein. 


 §.§ Riemannian Geometry and Affine Connection Systems


Let Q be an n-dimensional smooth manifold representing the configuration space of a simple mechanical system. Let TQ = _q∈ QT_qQ and T^*Q = _q∈ QT_q^*Q be the disjoint union of tangent and cotangent spaces called the tangent and cotangent bundle, respectively. Let π_TQ:TQ → Q be the bundle projection map for TQ and let TQ be the set be the set of vector fields on Q. In local coordinates (U, = (q^1,…,q^n)), we denote the coordinate induced basis by {q^1,…, q^n} for T_Q and { q^1,…, q^n} for T^*_Q. The natural pairing .|.:T^*_Q× T_Q → between vectors and covectors in the tangent and cotangent spaces, respectively, is defined by 

    _|_ = ∑^n_i=1_i X^i,   _∈ T^*_Q,  _∈ T_Q.
 where _ = ∑_iX^iq^i and _ = ∑_i_i q^i in the canonical basis of the tangent and cotangent bundles, having components X^1,…,X^n ∈ and _1,…,_n ∈, respectively.
 
Let _:T_Q × T_Q → be a Riemannian metric on Q defining the kinetic energy of the system <cit.>. For each ∈ Q the metric  induces two isomorphisms between the tangent and cotangent bundles given point-wise by 

    ♯_:T^*_Q→ T_Q
       _↦(_^♯_,_) = _|_,  ∀_∈ T_Q
        
          ♭_:T_Q→ T^*_Q
       _↦_q^♭_|_ = _(_,_),  ∀_∈ T_Q
 called the sharp and flat isomorphisms, respectively <cit.>. In the canonical local coordinates for TQ and T^*Q, the metric tensor takes the matrix form such that by abuse of notation we denote the matrix by _. As the result, the sharp and flat isomorphisms become _^♯_ = _^-1_ and ^♭__ = __, where _ and _ are expressed in the canonical bases. 

The Levi-Civita connection , is the unique affine connection associated with  and is derived from the assumption that the connection is divergentless and torsion-free <cit.>. The ∇_ be the covariant derivative of a vector field  along a vector field  with respect to ∇ is given by 

    = (Y^iq^jX^j + Γ^i_jkX^jY^k )q^i,
 where the Levi-Civita connection coefficients – also called the Chistoffel symbols– Γ^i_jk in the coordinate-induced basis is expressed as 

    Γ^i_jk = 1/2^il(_jlq^k + _klq^j - _jkq^l).
 The Christoffel symbols have the physical meaning for a simple mechanical system of Coriolis, centripetal and centrifugal accelerations <cit.>. The action of  on (1,1)-tensors ∈(T^*Q ⊗ TQ) (such as a projection map) is given by

    () = (^i_jq^kX^k + Γ^i_kl^l_jX^k - Γ^l_kj^i_lX^k)q^i⊗ q^i.
  We may consider the connection coefficients as the point-wise bilinear map Γ:T_Q× T_Q → T_Q as the function (_,_)↦Γ(,_)_. 

The Lagrangian L:TQ → for a simple mechanical system is defined as 

    L = 1/2_(_,_) - V().
 The Euler-Lagrange equations of motion for Lagrangian L are equivalent to forced affine connection system (∇,,V,F) 

    (t)(t) = (- V + F)^♯_,   t(t) = (t).
 defined by the connection and the gradient of the potential and external force F ∈ T^*Q <cit.>. 







  §.§.§ Nonholonomic Affine Connection System


We review the basic results of nonholonomic affine connection systems treated. For general references, see <cit.>.

Let {a^l⊂ T^*Q |  l=1,…, m} be a collection of everywhere linearly independent one-forms over Q such that they specify an (n-m)-dimensional nonholonomic distribution  at each point ∈ Q by 

    _ = {_∈ T_Q  |  a^l_|_ = 0,   l =1,…,m }.
 The constraint one-forms are locally given by a^l_ = ∑_i= 1^n^l_i() q^i for each l = 1,…,m. A basis for _ is chosen by selecting m everywhere linearly independent vector fields, {_1(),…,_n-m()} satisfying the constraint equation (<ref>). Using the Riemannian metric we complete the basis for T_Q by defining 

    ^_ := {(a^1_)^♯_,…, (a^p_)^♯_}.
 
Therefore, each tangent space can be -orthogonally decomposed into T_Q ≅_⊕^_. Define the projection maps _:T_Q →_ and ^_:T_Q →^_ at each point ∈ Q. The nonholonomic distribution  can be described in terms of the projection maps as

    = {∈ TQ   |   = (),  ∈ TQ} = {∈ TQ   |  ^() = }.

 To represent the evolution of the trajectory of the system :[a,b]→ Q, we form the nonholonomic affine connection system:

    _(t)(t) = (t) -( V )^♯_, 
       ∈^((t)) = , 
       t(t) = (t),
 where  is a section in ^ representing the Lagrange multipliers representing the constraint reaction forces given by

    (t) = - (_^)() + ^( V )^♯_.
 


    The nonholonomic affine connection <cit.> is obtained by substituting the Lagrange multipliers (<ref>) into the (<ref>) and restricting the equations of motion (<ref>)-(<ref>) to the distribution .








 §.§ Geometry of the Second-Tangent Bundle

In this section, we review the geometry of the second tangent bundle TTQ. First, we review the bundle structure and decomposition <cit.> which is required to compute the covariant derivative of a general nonlinear vector bundle map. The covariant derivative of a general nonlinear vector bundle map is a prerequisite for the main results of this article. It is noteworthy that this specific result isn't explicitly stated in the literature. Lastly, we review the second tangent bundle in the context of geometric mechanics <cit.>. For general references, see <cit.>. 



  §.§.§ Vertical/Horizontal Decomposition of TTQ

Consider the second tangent bundle of Q with bundle structure given by π_TTQ:TTQ → TQ. Local coordinates for T_(,_)TQ by (,_,_(,_),_(,_)) where (_(,_),_(,_)) given in the canonical coordinate induced basis by

    _(,_) = ∑_i=1^nV^iq^i |_(,_),   _(,_) = ∑_j=1^nW^jv^j |_(,_).
 where q^i :=π^*_TQq^i. The push-forward of the projection map π_TQ induces a bundle structure (π_TQ)_*:TTQ → TQ. In these coordinates, the bundle projection maps satisfy 

    π_TTQ: (,,,) ↦(,)
       (π_TQ)_*: (,,,) ↦(,).
 The vertical bundle is defined by TQ = (π_TQ)_* and given locally by 

    TQ    = {(,,,) ∈ TTQ  |  (π_TQ)_*(,,,) = }
       = {v^i |_(,_)  |   i = 1,…,n }.
 
The parallel transport equation induces a vector-valued connection one form ω∈Ω^1(TQ;End(^n)) given by (see section 9.7 of <cit.>)

    ω :=  v^k + Γ^k_ijv^j x^i.
 Define the horizontal distribution TQ = (ω) ⊂ TTQ  <cit.> which is given locally by 

    TQ = i =1,…,n{q^i |_(,_) - Γ^k_ijv^jv^k |_(,_)}.
 Therefore, the second tangent bundle is decomposed as TTQ ≅ TQ≅ TQ. Define the vertical lift <cit.> of a vector field ∈ TQ to the vertical space _(,_)TQ

    : TQ →_(,_)TQ
       ↦t(_ + t) |_t=0 .
 In coordinates, the vertical lift is given by ^ = X^iv^i. The vertical lift is a smooth bijection and hence provides an isomorphism between the _(,_)TQ and TQ given by := ()^-1

    :_(,_)TQ →  TQ
       (,_,,) ↦(,).
 Similarly, we define the horizontal lift <cit.> with respect to the affine connection  with Christoffel Symbols Γ by

    : T_qQ× TQ →_(,_)TQ
       (,_,) ↦(,_,,- Γ(,_)).
 and its inverse := ()^-1 given by the map

    :  _(,_)TQ → TQ
       (,_,, - Γ(,_)) ↦(,).
 Lastly, we define projection maps from the tangent of the tangent bundle onto its vertical and horizontal subbundles. The projection onto the vertical bundle is defined intrinsically by ^_Γ = id_ TQ⊕Γ and given locally by 

    ^_Γ: TTQ → TQ
       (,_,,) ↦(,_,,  + Γ(,_)).
 From the projection property, the projection onto the horizontal space TQ is defined intrinsically as _Γ^:= id_TTQ - _Γ^ and given locally by 

    _Γ^: TTQ→_(,_)TQ
       (,_,,) →(,_,,-Γ(,_)).
 From these projection maps, we define projection from TTQ to two copies of TQ, each of which is isomorphic to TQ and TQ, respectively. These projections are defined in the following diagram:


    

    TTQ       𝒱TQ       TQ 
    
    	TTQ       ℋ TQ       TQ
    	["𝒬^𝒱_Γ"', from=1-1, to=1-3]
    	["↓_ver"', from=1-3, to=1-5]
    	["_𝒱", curve=height=-18pt, from=1-1, to=1-5]
    	["𝒬^ℋ_Γ", from=2-1, to=2-3]
    	["↓_hor", from=2-3, to=2-5]
    	["_ℋ"', curve=height=18pt, from=2-1, to=2-5]


 Therefore, the isomorphism TTQ ≅π_TQ^*TQ⊕π_TQ^*TQ is established by _⊕_. 

    The projection maps _:TTQ → TQ and _:TTQ → TQ satisfy the following properties: 
    
        
  * _  = (π_TQ)_*
        
  * _(_*) =, for a section :Q → TQ.
    


    See appendix <ref>



  §.§.§ Covariant Derivative of a Tangent Bundle Map

In this subsection, we find the covariant derivative of a general (nonlinear) tangent bundle map. This covariant derivative is essential for the results in subsequent sections. The authors have not found this result explicitly stated in the literature. Lastly, we provide a matrix formulation of the equations for practical computations.  

 
Let :TQ → TQ be a general (not necessarily linear) vector bundle map over the tangent bundle π_TQ:TQ → Q. Let , :Q → TQ be a sections. Then, the covariant derivative of ∘ along  is     
    (∘) =  ∇^_() + D^(),
 where the vertical Jacobian D^ = _D |_ TQ is given locally by 

    D^() = ^mw^l()^l q^m,
 and the horizontal connection () = _D|_ TQ is given locally by 

    () = (^mq^kv^k  + Γ^m_kiv^k^i - Γ^l_kiw^kv^i^mw^l)q^m.



    See appendix <ref>




    Note that when the bundle map :TQ → TQ linear in velocities, the vertical derivative D^ = is linear in its argument, and the horizontal covariant derivative  reduces to the usual covariant derivative  e.g., see equation (<ref>). Therefore, this "chain rule" is a generalization of the traditional chain rule for the covariant derivative of a linear vector bundle map. 



    For practical calculations, in the coordinate induced basis {q^1,…,q^n}, the covariant derivative  given in (<ref>) may be calculated in matrix form as
    
    = [] + [(,)],
 where [] = [Y^iq^k] is the Jacobian of  and the Christoffel symbols are organized in a matrix [(,)] = [Γ^i_kjX^j]. Similarly, the covariant derivative of a (1,1)-tensor  given in (<ref>) may be calculated in matrix form as
    
    [] = [∑^n_k=1 q^kX^k] + [(,)][] - [][(,)].
 Lastly, the covariant derivative  of a tangent bundle map (,) as in (<ref>) may be calculated in matrix form by 
    
    = [(,)] + [(,)] - [D^][(,)].





  §.§.§ Dynamics of Mechanical Systems on TTQ

A vector field :TQ → TTQ is a second order system on Q if and only if for each integral curve :[a,b]→ TQ of  we have t(π_TQ∘)(t) =  (π_TQ)_* = (t), summarized in the following diagram <cit.>




    TTQ 
    
    	TQ        TQ       
        Q
    	["(π_TQ)_*"'pos=0.2, from=1-2, to=2-1]
    	["π_TTQ"pos=0.3, from=1-2, to=2-3]
    	["π_TQ", from=2-3, to=3-2]
    	["π_TQ"', from=2-1, to=3-2]
    	["X"', curve=height=30pt, from=2-3, to=1-2]
    	["X", curve=height=-30pt, from=2-1, to=1-2]
    	["id_TQ"', from=2-1, to=2-3]


 
If = (,,,) is a second-order vector field, then it follows that the velocity components satisfy v^i = V^i. 

Given, the decomposition TTQ ≅ TQ ⊕ TQ, we construct the vector field associated with the affine connection system (<ref>). Let (,):[a,b] → TQ be a curve of satisfying (<ref>). The forces acting on the system (- V^♯_ +F^♯_) in the right-hand side of (<ref>) are vertically lifted to _(,)TQ by (<ref>). The vector field associated with the parallel transport equation (t)(t) = on the left-hand side of (<ref>) is given by the horizontal lift (<ref>) of the velocity  to _(,_)TQ.
Therefore, the time variation of the curve t(,) ∈ T_(,)TQ ≅_(,)TQ⊕_(,)TQ satisfies the differential equations 


    t = -Γ^k_ijv^iv^jv^k + ( - Vq^i + F_i)^ikv^k,
       t = v^iq^i.
 

    When there are no external forces in (<ref>), the vector field :TQ → TTQ
    
    (,) = v^iq^i - Γ^k_ijv^iv^jv^k,
 is called the geodesic spray <cit.>. The integral curves of the geodesic spray  in TQ project to geodesics with respect to  in Q.










§ AFFINE CONNECTION SYSTEM AS A SLOW/FAST SYSTEM WITH STRONG FRICTION
  

Let Q be a smooth manifold and let ⊂ TQ be a linear subbundle representing a set of m ideal nonholonomic constraints. To realize the nonholonomic constraints by friction forces, consider a Rayleigh dissipation function :T_qQ× T_qQ →, represented as a symmetric (0,2)-tensor on each tangent space <cit.>. This function induces a dissipative friction force given by the (negative) fibre derivative -:TQ → T^*Q of the dissipation function. Assume that this friction force satisfies = () to ensure that the nonholonomic distribution is attractive. 

We construct this dissipation function   as follows. Consider a ^m-valued one-form a_:T_Q →^m consisting of the m nonholonomic constraint one-forms a^l∈ T^*Q for l = 1,…,m. For each configuration ∈ Q, define the friction coefficients as the symmetric linear vector bundle map μ̅(): ^m→^m*≅^m and finally, define the adjoint of the map representing the constraint as a^*_:^m*→ T^*_Q. Define the Rayleigh dissipation function = 1/2a^*_∘μ(q) ∘ a_ and in local coordinates is given by 

    = 1/2_i^l()μ̅_lk()^k_j()__ij() q^j⊗ q^i.

The dissipative Rayleigh friction force with large coefficients is defined as follows. Fix a small positive real number >0 (which is thought to be the time scale on which this force acts) and rescale the friction coefficients μ̅ = 1/μ to find the friction force given by 

    F_ = -1/() = - 1/_ij()v^i q^j∈ T^*Q.
 Let :[a,b] → Q be a smooth curve that satisfies the equations of motion formulated by the forced affine connection system 

    = -( V )^♯_  - 1/()^♯_,
       t = .



 As this is a singularly perturbed system, it follows that (<ref>-<ref> emits a normally hyperbolic invariant manifold (NHIM) for >0 <cit.>. That is, for small values of , there exists a perturbed manifold _ that is C^1-close and diffeomorphic to the nonholonomic constraint distribution . The existence of this submanifold has been studied by Eldering and summarize the result <cit.> as follows. Write the system as a vector field (<ref>) on TQ according to the decomposition TTQ ≅ TQ ⊕ TQ: 
 
    t(,) = (,) + (- V^♯_)^ - (1/()^♯_)^
 where we have vertically lifted the forces to TTQ using (<ref>). Make the reparameterization of the trajectory to fast-time τ = t/ to find  
 
    τ(,) = (,) +(- V^♯_+ U^♯_)^ -  (()^♯_)^.
 When = 0 then the fast time system is of the form 
 
    τ(,) = - (()^♯_)^.
 When this friction force acts on an instantaneous time-scale (= 0) it is the dominant force acting on the system. By construction, when we restrict the dynamics to the distribution , the friction force vanishes; thus  is an invariant manifold for the fast-time dynamics (<ref>). When the dynamics are restricted to the -orthogonal complement ^, we find that the dynamics are attractive. Therefore, by the theory of normally hyperbolic invariant manifolds <cit.> there exists a perturbed manifold _ that is C^1-close and diffeomorphic to .

 

 §.§ Representation of the Slow Manifold

 We represent the perturbed manifold _ as the image of a section _:→ TQ over the bundle :TQ →. By definition, the section _ satisfies the following commutative diagram














    𝒟       TQ 
       𝒟["H̃_ε", from=1-1, to=1-3]
    	["𝒫", from=1-3, to=2-2]
    	["id_𝒟"', from=1-1, to=2-2]


 That is, ∘_ = id_. By the projection property id_TQ = ⊕^, we find that the section _ is given by 
 
    _ = id_ + ^∘__=:_
 where _:→^ is a section over π_TQ:TQ → Q. Therefore, the slow manifold _ is defined by the set 
 
    _ = {∈ TQ   |   = ^ + _(^),  ^∈}.

 
 
 Using the projection maps, we then define the sections _ = _∘:TQ→ TQ as 
 
    _() = () + _∘(),
 where _∘ = ^∘_∘:TQ →^. Therefore, the perturbed manifold _ = (_) is given by 
 
    _ = {∈ TQ  |   = _() = () + _∘(),  ∈ TQ }.
 Note that the form of this function _ depends on the choice of projection map . 
 
     Physically speaking, the slow manifold represents the slip velocities in the orthogonal complement ^ and are a function of the variables in the nonholonomic distribution . 
 


    The map _:TQ → TQ for the system (<ref>)-(<ref>) is a nonlinear projection onto _. 


    Given a section :Q → TQ we find
     
    _∘_()    = _(() + _∘() )
       = (_∘)(() + _∘() )
       = _(∘() + ∘_∘() )
       = _(∘() + ∘^∘_∘() )
        = _∘() = _().

  

    Since _ defines a nonlinear projection map, we may also define its complement ^_ := id_Q - _. This defines a projection map onto the complement of the slow manifold abbreviated by TQ/_. 

 The map ^_:TQ→ TQ/_ is a nonlinear projection map that is complementary to _.


    First, we show that _ is a projection map. Let :Q→ TQ be a section. Then, 
    
    _^∘_^()   =(id_TQ - _)∘( - _())
       = - _() - _() + _∘_()
       =  - _() = _^().
 Lastly, we show that this map is complement to _.
    
    _∘_^() = _() - _∘_() = 0,
       _^∘_() = _∘_() - _()   = 0.



    As a consequence of the complementary projection map in Proposition <ref>, we find that ∈_ if and only if 
    
    _^()    =  -_() =  - () - _∘()
       = ^() - _∘() = 0.
 This is summarized in the following proposition.  


  
     ∈_ if and only if ^() = _∘().
 
  Suppose that ∈_. Then, there exists ∈ TQ such that = () + _(). We calculate
     
    ^()    = ^(() + ^∘_())
       = ^∘^∘_() = _() = ^∘_∘().
 Moreover, we find that  and  project to the same vector
 
    () = (() + _∘() )) = ().
 Therefore, ^() = ^∘_∘() = _∘().
 
 Conversely, suppose ∈ TQ such that ^() = _∘(). Then, since  is a projection map, we have that _TQ = ⊕^. Therefore, since we assume that ^() = _∘(), we find
 
    = () + ^() = () + _∘().
 Therefore, ∈_.
  By Proposition <ref>, we find that the distribution can be represented as the set

    _ = {∈ TQ   |  ^() = _∘() }.

 

    The equations of motion for a curve :[a,b] → Q lying on the slow manifold _ are given by

    (t)(t) = - V^♯_  - 1/((t))^♯_
       t(t) = (t),
       ^((t)) = _∘((t)) (t) ∈_.
 


    The equations of motion (<ref>)-(<ref>) are described by a forced affine connection system. The existence of the slow manifold follows from the geometric singular perturbation theory of normally hyperbolic manifolds <cit.>. By Proposition <ref> the slow manifold is represented by (<ref>).

















§ APPROXIMATION OF THE SLOW SUBSYSTEM
 

     In this section, we find a generating equation for the invariant manifold _ using the equations of motion (<ref>) - (<ref>).  The strategy is to consider velocities restricted to the slow manifold _ and expand the section _ as a power series and match terms to the desired order in ϵ. 



 §.§ Generating equation for the slow-manifold

Let us find an expression for the dissipation function ()^♯_ in terms of the unknown function _ describing the perturbation of the ideal constraints. Using the projection maps, we project the dynamical equations (<ref>) onto ⊕^≅ TQ

    () = -( V^♯_)
       ^() = -^( V^♯_) - 1/^♯_().
 


Consider the transformation ^♯__:T_Q → T_Q, then there exists a transformation _:^_→^_ satisfying _∘^♯__ = ^_.
 

    The friction force is a linear transformation ^♯__:T_Q → T_Q satisfying = (^♯__) and im(^♯__) = ^. Thus, by the first isomorphism theorem <cit.>, we have the result.











 
 
    The map _ may be constructed as follows. Let Φ() be a frame spanning _⊕^_. Define linear maps in this frame with the subscript Φ, so, [_Φ] = Φ^-1[]Φ, [^_Φ] = Φ^-1[^]Φ and [_Φ^♯_] = Φ^-1[^♯_]Φ and the equation [_Φ^]=[_Φ][_Φ^♯_] is given by the block matrix equation of the form
    
    [  ;   ]_[^_]  = [  ;   ]_[_Φ][  ;   ]_[_Φ^♯_].
 Therefore, we set = ^-1 and perform the inverse transformation [_] = Φ[_Φ]Φ^-1 to obtain the result.


 
    Let _:→ TQ be the section defining the invariant manifold _ with respect to the dynamics (<ref>)-(<ref>). Let _ = ∑_k ≥ 0^k^(k) be an expansion of the section by a formal power series. Then, the slow manifold may be recursively approximated from the following generating equation 
    
    _(^) = _∘[(^^)(^) - ^( V^♯_) ] 
        + _∘[ (^^)(_) + (_^)(^) + (_^)(_) ] 
       - _∘ D^_[(^)(^)  - ( V^♯_) ] 
        - _∘ D^_[(_)(^)  + (^)(_) + (_)(_)  ] 
              - _∘[__(^) + __(^) ].




    By analogy with the ideal nonholonomic case <cit.>, take the covariant derivative of the latter invariant manifold equation (<ref>) 

    (^()) = (_∘())
 and substitute the affine connection system (<ref>)-(<ref>). The left-hand side is given by 

    (^() )   = (_^)() + ^(_)
       = (_^)()  -1/()^♯_ - ^( V^♯_).
 On the other hand, according to Proposition <ref> and the chain rule, we find the right-hand side 

    (_∘() )= D^_((()))  + _(())
        = D^_(()() + () )  + _(())
        = D^_(()() )-D^_(( V^♯_))  + _(()).
 Putting both sides of this calculation together, we find the following identity for the dissipative forces 

    -1/^♯_() = -(^)() + ^( V^♯_)
       + D^_(()() ) + _(()) - D^_(( V^♯_))
  

Lastly, we restrict (<ref>) to the velocities ∈_ to obtain the generating equation for _

    ^♯__(^)   = (_^ + _^)(^ + _) + ^( V^♯_)
        -  D^_(- V^♯_)  - (_^ + __)(^) 
        + D^_( (^ + _)(^ + _) ).
 By Lemma <ref> and linearity we obtain the generating equation (<ref>).


    Note that there is an  in all terms on the right-hand side of (<ref>). Thus, any power series expansion of _ results in an order higher on the right-hand side than the left-hand side. Therefore, the generating equation (<ref>) provides a recurrence relation for the approximation of _.

 
  The generating equation for _(^) in (<ref>) results in the same reaction forces -1/^♯__(^) as those reaction forces resulting from the classical invariance condition 

    t^() |__ = t_∘() |__.



    Without loss of generality, assume that the potential V=0. The equations of motion (<ref>)-(<ref>) can be written as 
    
    = -(,) - 1/^♯_()
       =∈_^() = _∘().
 Taking a Lie derivative of (<ref>) along the along these dynamics we find 
    
    [∑^n_k=1^q^kv^k] - [^][(,)] - 1/[^][^♯_] = 
       [_(,())] + [D^_][∑^n_k=1q^kv^k]
          - [D^_][][(,)] - 1/[D^_][][^♯_].
 where the quantities in the square brackets are the compact matrix notation introduced in Remark <ref>. As = ^♯_, the last term vanishes. Let = ^_∈_ be a velocity in the slow manifold, then the reaction forces are restricted to _ are given by 
    
    -1/^♯_(^_) = -[∑^n_k=1^q^kv^k__]^_ +[^][(,^_)]^_
            +[_(,^)]^_ + [D^_][∑^n_k=1q^kv^k__]^_- [D^_][] [(,^_)]^_.
 Contrastingly, the reaction forces _ resulting from the generating equation (<ref>) are calculated from the equation
    
    -1/^♯_() = -(^)() + D^_()() + _(()).
 as was observed in equation (<ref>). By Remark <ref> the covariant derivatives in this equation may be expressed in matrix form by 
    
    -1/^♯_() = -[∑^n_k=1^q^kv^k] - [(,)][^] + [^][(,)]
        + [D^_][∑^n_k=1q^kv^k] + D^_[(,)][] -[D^_][][(,)]
        + [_(,^)] + [(,)]_(,^)- [D^_][(,)][].
 Then, restricting to = ^_, and using ^∘_ = _ terms cancel and therefore, these friction forces (<ref>) restricted to _ are identical to (<ref>).


 The dynamics defined by (<ref>)-(<ref>) restricted to _ are tangent to the manifold _.


    In local coordinates (<ref>)-(<ref>) are equivalent to the equations of motion on TTQ given by (<ref>). Following singular perturbation theory, the classical invariance equation (<ref>) ensures that the reaction forces result in dynamics tangent to _.
    By Theorem <ref>, the reaction friction forces described by (<ref>) are equivalent to the reaction forces resulting from the classical invariance equation. Therefore, the dynamics (<ref>)-(<ref>) along with the generating equation (<ref>) result in dynamics tangent to _. 



    The classical invariance condition (<ref>) for the dynamics (<ref>)-(<ref>)
is equivalent to the condition 

    (^())|__ = (_∘())|__.



    By Theorem <ref> and Corollary <ref> both conditions result in the same dynamical equations of motion restricted to the slow manifold _. 


    The later covariant derivative invariance condition (<ref>) may be geometrically interpreted as the parallel transport of the condition ^() = _∘() along trajectories = ∈_ lying in the slow manifold.






 §.§ Approximation of Slip Velocities

In this section, we employ the generating equation (<ref>) from Theorem <ref> to compute the zeroth, first and second order approximations of the slow manifold _. Expand _ as a formal power series 

    _ = ^(0) + ^(1) + ^2^(2) +  O(^3).
 Further expanding the power series allows for higher order approximations of _.


  §.§.§ Zeroth Order Approximation of Slip Velocities
 Lets consider the zeroth order term of the slow manifold _ = ^(0) + O(). Observe that when = 0 we have 

    _ |_ = 0 = _0  = {∈ TQ   |  ^() = ^(0)}.
 Since _ is a perturbation of the nonholonomic distribution, we find that ^(0) =. Then, to find the zeroth order velocity,  solve the identity ^() = for = ^∈.

    The zeroth order approximation of the slow fast velocities are equivalent to the nonholonomic velocities.  





  §.§.§ First Order Approximation of Slip Velocities
 Let us assume that section _ is taken to first order, so we expand _ = ^(1) + O(^2). Then according to the generating equation (<ref>) to this first order slip velocity, we find 

    ^(1) = _∘[(^^)(^) - ^( V^♯_)].



    The first order approximation of the slow fast velocities are proportional to the nonholonomic Lagrange multipliers (<ref>). This is consistent with previous results <cit.>.





  §.§.§ Second Order Approximation of Slip Velocities
 Lastly, assume that section _ is taken to second order, so we expand the formal power series as _ = ^(1) + ^2^(2) + O(^3). Then according to the generating equation (<ref>) to this first order slip velocity, we find 

    ^(2) = _∘[(^(1)^)(^) + (^^)(^(1))] 
              -_∘[D^^(1)((^)(^) ) - D^^(1) ( V^♯_) ]- _∘[^^(1)(^) ].



    Higher order approximations of _ may be calculated recursively in terms of higher order covariant derivatives ^n. However, in practice, first and second order approximations suffice. This is due to the fact that the friction forces - 1/^♯_() in (<ref>) restricted to the second order approximation results in a first order force - ^♯_(^(2)) that is a function of zeroth and first-order velocities. 





 §.§ Approximation of Slip Dynamics

In this section, we use the approximations of the slip velocities _ = ^(1) + ^(2) + O() to approximate the zeroth and first order dynamics. Due to the recurrence relation defined by the generating equation (<ref>) on _ we observe that the zeroth order approximation of the dynamics (<ref>)-(<ref>) depends on the first order approximation ^(1). Furthermore, the first order approximation of the dynamics (<ref>)-(<ref>) depends on the second order approximation ^(2), and so on. Therefore, the order of the slip velocities _ is always one order above the order of the dynamics. 



  §.§.§ Zeroth Order Dynamics
 By the equations of motion (<ref>) and the first order approximation of the slow manifold _ = ^(1) + O(^2), we find the zeroth order equations of motion are given by

    t = -Γ(,^)^ - (^^)(^) - ( V^♯_)
       t  = ^∈_0≅.



    The zeroth order equations of motion (<ref>) are exactly the nonholonomic equations of motion (<ref>)-(<ref>). This result is consistent with previous realizations of slow/fast nonholonomic systems <cit.>. 





  §.§.§ First Order Dynamics
 Using the generating function (<ref>) we obtain the first and second order approximations _ = ^(1) + ^2^(2) + O(^3) to obtain (<ref>) and (<ref>), the former gives the slip velocity and the later gives the reaction force. Restricting the dynamics to the first order approximation of slip, we find the dynamics: 


    t   = 
        -Γ(,^)^ - (^^)(^) - ( V^♯_)
          -[Γ(,^)^(1) +Γ(,^(1))^] +^^(1)(^)
          -[(^^)(^(1)) + (^(1)^)(^) ]
           +   D^^(1)[((^)(^) ) - ( V^♯_) ]
        
       t = ^ + ^(1)(^)
 where the first order approximation of the slip ^(1) is given by (<ref>), and vertical derivative D^^(1) and horizontal ^^(1) is calculated according to equation (<ref>) and (<ref>). 



    











§ CASE STUDY: VERTICAL ROLLING DISK
 
To illustrate the theory, we consider a vertical rolling disk for simplicity. However, our approach may be applied to other systems with (linear) nonholonomic rolling constraints such as wheeled mobile robots <cit.> and vehicle systems <cit.>. For the traditional derivation of the nonholonomic equations for the vertical rolling disk using the Lagrange-d'Alembert variational principle see <cit.> and for the affine-connection approach see <cit.>, and <cit.>. 

The configuration space for a planar vertical rolling disk is Q = SE(2)× SO(2) with local coordinates = (θ,x,y,φ) and local coordinates for velocities = (v_θ,v_x,v_y,v_φ). The Riemannian metric defining the kinetic energy is given by 

    _ = diag(I,m,m,J )
 where m is the mass of the disk, I is the moment of inertia normal to the plane and J is the moment of inertia about the rolling axis. Note that as _ is constant, the corresponding Christoffel symbols (,) = for all (,) ∈ TQ , the metric _ is flat. 

The nonholonomic constraint one-forms 

    a_q^1 =  x - Rcosθφ,   a_q^2 =  y - Rsinθφ,
 enforcing rolling without slipping are given by 

    a_q^1(v_θ,v_x,v_y,v_φ) = v_x - Rcosθ v_φ = 0,
       a_q^2(v_θ,v_x,v_y,v_φ) = v_y - Rsinθ v_φ = 0,
 where R is the radius of the disk. This constraint defines a nonholonomic distribution _ = (a^1_q)∩(a^2_q) which are spanned by the vector fields 

    θ,    Rcosθx + Rsinθy + φ.
 Using the musical isomorphisms, we find that the -orthogonal complement spanned by the orthogonal complement ^ = (^∘)^♭_ is spanned by the vector fields 

    1/mx -R/Jcosθφ,   1/my - R/Jsinθφ.
 The projection map :TQ → onto the distribution  is constructed by the vector fields spanned by (<ref>) and the metric (<ref>) and is given by 

    = [       1/        0        0        0;        0   cos^2θ 1/2sin2θ J/mRcosθ;        0 1/2sin2θ   sin^2θ J/mRsinθ;        0   cosθ/R   sinθ/R   J/mR^2 ]
 where := mR^2/J+mR^2 is a ratio of inertia. 

Similarly, the projection map ^:TQ →^ onto the distribution ^ is constructed by the vector fields spanned by (<ref>) and the metric (<ref>) and is given by 

    ^ = [                   0                   0                   0                   0;                   0 J + mR^2sin^2θ/mR^2           -1/2sin2θ           -J/mRcosθ;                   0           -1/2sin2θ J + mR^2cos^2θ/mR^2           -J/mRsinθ;                   0             -cosθ/R             -sinθ/R                   1 ].
 The nonholonomic constraints may then be expressed by the equation ^() =, from which we find the constrained velocities ^∈ is given by 

    ^ = [       v_θ Rcosθ v_φ Rsinθ v_φ       v_φ ]^T
 in the coordinate induced frame {θ,x,y,φ}.

We construct the Rayleigh dissipation function according to equations (<ref>) using the nonholonomic constraint form (<ref>), from which the coefficient matrix is given by 

    ^♯_ = [          0          0          0          0;          0        μ/m          0 -μ R/mcosθ;          0          0        μ/m -μ R/msinθ;          0 -μ R/Jcosθ -μ R/Jsinθ    μ R^2/J ]
 where the units of the friction coefficients μ are [kg/s ], and the corresponding accelerations given by _ = -1/^♯_. Defining a frame Φ as the direct sum of the vector fields (<ref>) and (<ref>) we find the matrix _ according to Remark <ref> is given by 

    _ = [                    0                    0                    0                    0;                    0 J^2 + γ m R^2 sin^2θ      -1/2γ mR^2sin2θ             J^2Rcosθ;                    0      -1/2γ mR^2sin2θ J^2 + γ m R^2 cos^2θ          - J^2 Rsinθ;                    0             -mRJcosθ             -mRJsinθ                mR^2J ]
 where γ = mR^2 + 2J. One can check that _ satisfies Lemma <ref>. As the Christoffel symbols are all zero, the equations of motion are simply given by 


    t[ v_θ; v_x; v_y; v_φ ]  = -1/[                                        0;                    μ/mv_v - μρ/mcosθ v_φ;                    μ/mv_y - μρ/msinθ v_φ; -μρ/Jcosθ v_x - μρ/Jsinθ v_y + μρ^2/Jv_φ ],   
       t[ θ; x; y; φ ] = [ v_θ; v_x; v_y; v_φ ].


    Note that the dynamics of the orientation in the plane θ is of the form v̇_θ = 0 as v_θ∈ and = (^♯_), by construction. Hence, the orientation in the plane is invariant for the dynamics of the vertical rolling disk. 




 §.§ Approximations of Slip Velocities
 The first order approximation of the slip velocities are calculated according to  (<ref>) and are found as 

    ^(1) = _∘(^^)(^) =  [                0;  mR/μsinθ v_θv_φ; -mR/μcosθ v_θv_φ;                0 ]
 Further, the second order approximation of the slip velocities is calculated according to (<ref>) using the results of Remark <ref> and is found to be 

    ^(2)   =     _∘[(^(1)^)(^) + (^^)(^(1))] 
              -_∘[D^^(1)((^)(^) ) ] - _∘[^^(1)(^) ] 
       =m^2RJ^2/μ^2(J + mR^2)^2[              0; cosθ v_θ^2 v_φ; sinθ v_θ^2 v_φ; -mR/Jv_θ^2 v_φ ]
 




 §.§ Approximations of Slip Dynamics
 Using the first order approximation of the slip velocities (<ref>), we find the zeroth order approximation of (<ref>)-(<ref>) using equations (<ref>) are given by 

    t[ v_θ; v_x; v_y; v_φ ] = [              0; -Rsinθ v_θ v_φ;  Rcosθ v_θ v_ϕ;              0 ] +O(), 
       t[ θ; x; y; φ ] = [       v_θ; Rcosθ v_φ; Rsinθ v_ϕ;       v_φ ] +O().
 


    The zeroth order equations of motion (<ref>)-(<ref>) are the same equations of motion resulting from Lagrange-d'Alembert's variational principle <cit.>. 

Moreover, using the second order approximation of the slip velocities (<ref>) and (<ref>), we find the first order equations (<ref>) are given by 

    t[ v_θ; v_x; v_y; v_φ ] = [              0; -Rsinθ v_θ v_φ;  Rcosθ v_θ v_φ;              0 ] +mRJ/μ(J+mR^2)[              0; -cosθ v_φv_θ^2; -sinθ v_φv_θ^2;  mR/J v_φv_θ^2 ]+O(^2),
       t[ θ; x; y; φ ] = [       v_θ; Rcosθ v_φ; Rsinθ v_φ;       v_φ ] + mR/μ[            0;  sinθ v_θv_φ; -cosθ v_θv_φ;            0 ] +O(^2).
 




§ CONCLUSION

In this paper, we developed an affine connection approach for realizing nonholonomic mechanical systems. We formulated the slow/fast dynamics in a coordinate-independent manner in terms of sections, projection maps and covariant derivatives. We proposed a novel invariance condition for slow/fast realizations of nonholonomic constraints and proved that this novel condition is equivalent to the classical approach. We proposed a recursive procedure to approximate higher order slip velocities and their associated dynamics. We conclude the paper with some remarks regarding the advantages of the proposed formalism. 

    
  * The geometric formalism is coordinate-independent and suitable for symbolic computation.
    
  * The proposed slow-fast decomposition permits a recursive procedure to approximate the invariant manifold representing slip velocities. In the context of vehicle systems, the slow manifold may be physically interpreted as a deviation from the ideal no-slip constraints. Approximations of the slow manifold provide estimates of these deviations and have applications in model-based slip estimation and localization in vehicle systems. 
    
  * Higher order approximations of slip velocities are based on objects that can be interpreted geometrically and may give insight into the geometric structure of other equivalent approaches. 
 The future directions of this research include the following: (i) Designing a recursive controller robust to the uncertainties in the estimation of the friction coefficients and external forces; (ii) Use of higher-order approximations of the slow-manifold for in-the-loop motion estimation of vehicle systems; and (iii) symmetry reduction for slow-fast realizations of nonholonomic systems. 







Conflict of Interest:  The authors declare that they have no conflict of interest.





§ PROOFS OF STATEMENTS




 §.§ Proof of Proposition <ref>
 
 
    For the horizontal space  

    _(,_,,)    = (_Γ^(,_,,))^
       = (,_,,- Γ(,_))^
       = (,) = (π_TQ)_*(,_,,).
 Therefore we have _ = (π_TQ)_*. 

For the vertical space, let :Q → TQ be a section:

    _(_*)    = (^_Γ(,_,_,D()·_) )^
       = (,_,, D()·_ + Γ(,_)·_)^
       = (,D()·_ + Γ(,_)·_)
        = .







 §.§ Proof of Proposition <ref>
 












Let :TQ → TQ be a vector bundle map, i.e., π_TQ∘ = π_TQ. Let , :Q→ TQ be sections, then ∘ : Q → TQ is also a section. The push-forward of this section is a map that satisfies the following commutative diagram: 




    TTQ        TTQ 
    
    	TQ        TQ 
        Q
    	["h", from=2-1, to=2-3]
    	["π_TQ", from=2-3, to=3-2]
    	["π_TQ"', from=2-1, to=3-2]
    	["w", curve=height=-24pt, from=3-2, to=2-1]
    	["π_TTQ", from=1-3, to=2-3]
    	["h_*", from=1-1, to=1-3]
    	["w_*", curve=height=-12pt, from=2-1, to=1-1]
    	["π_TTQ", curve=height=-6pt, from=1-1, to=2-1]


 By virtue of Proposition (<ref>), we calculate the covariant derivative along  as follows

    (∘)   = _(D(∘) ) = _(D· D)
       = _(D· (_⊕_)(D)  )
       = _(D·(_(D)+_(D)  ))
       = _(D·(_(D)⊕) ) + _(D·(⊕) ),
 where we used the direct product isomorphism, twice and D is the Jacobian of . In the decomposition TTQ ≅ TQ⊕ TQ, consider the Jacobian of the map :

    D = [ _D|_ _D|_; _D|_ _D|_ ].
 Since _ = (π_TQ)_*,  when restrict D to the vertical space TQ = (π_TQ)_* we find 

    π_D|_ = (π_TQ)_*_*|_  = (π_TQ)_*|_  = .
 Let :[a,b] → TQ be a horizontal curve along (t) in Q. Then, ∈ T_(t)TQ ≅_(t)TQ⊕_(t)TQ satisfying 
    = ^i(q^i - Γ^k_ijv^jv^i).
 Since ((t)) ∈ T_(t)Q is a curve, we calculate derivative 

    (t)    = t(π_TQ∘∘(t)  ) = (π_TQ)_*  D  = _(D ).
 Therefore, we find that _D|_ is the identity map id. 

Define the vertical component of the Jacobian by D^:= _(D)|_ and the horizontal covariant derivative ∇^:= _D|_. 
Therefore, the covariant derivative of the bundle map  is given by 

    (∘) = () + D^().

We calculate local coordinate expressions for the covariant derivative as follows:

    (∘)   = v^iq^i((∘)^jq^j)
       = v^i (_q^i((,)^j)q^j + (,)^jq^iq^j)
       = v^i (^jq^iq^j + ^jw^kw^kq^iq^j + ^iΓ^k_ijq^k).
 On the other hand we have that 

    q^i = (w^kq^i + Γ_ijw^j)q^k.
 Putting this expression into the above formula we find 

    (∘)   = ^mw^l()^l q^m + (^mq^kv^k  + Γ^m_kiv^k^i - Γ^l_kiw^kv^i^mw^l)q^m
       =  D^() + ().
 




















 Paper Title: Infinitely many new renormalization group flows between Virasoro minimal models from non-invertible symmetries.






YITP-24-92











 Infinitely many new renormalization group flows between Virasoro minimal models from  non-invertible symmetries



Takahiro Tanaka and Yu Nakayama









Yukawa Institute for Theoretical Physics,
Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan







§ ABSTRACT 
Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models  ℳ(kq + I, q) →ℳ(kq-I, q) induced by ϕ_(1,2k+1). They vastly generalize the previously proposed ones k=I=1 by Zamolodchikov, k=1, I>1 by Ahn and Lässig, and k=2  by Dorey et al. All the other ℤ_2 preserving renormalization group flows sporadically known in the literature (e.g. ℳ(10,3) →ℳ(8,3) studied by Klebanov et al) fall into our proposal (e.g. k=3, I=1). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the SU(2)_q-2 fusion ring.


empty 












§ INTRODUCTION

While we believe we know everything about the Virasoro minimal models in two-dimensional conformal field theories, the renormalization group flow between them, in particular when one of them is non-unitary has been largely unknown. The Virasoro minimal model ℳ(p,q) is characterized by two coprime integers p and q, and we can compute all the conformal data for an arbitrary choice of (p,q). It has been, however, a surprisingly difficult question to ask if they are connected by a renormalization group flow, given two randomly chosen integers (p,q) and (p',q'), Our goal is to answer this question by using the non-invertible symmetries to classify the flows. Indeed, we will show that non-invertible symmetries give rise to infinitely many new renormalization group flows between two Virasoro minimal models.

Our infinitely many new flows
 ℳ(kq + I, q) →ℳ(kq-I, q) induced by ϕ_(1, 2k + 1) vastly generalize the previously proposed ones k=I=1 by Zamolodchikov <cit.>, k=1, I>1 by Ahn <cit.> and Lässig <cit.>, and k=2 by Dorey et al <cit.>.[Dorey et al also studied the flows corresponding to k=1/2 with odd q <cit.>. Since the preserved non-invertible symmetries become smaller, we study the generalization of half-integer k separately in section <ref>.] It also encodes the other ℤ_2 preserving renormalization group flow sporadically known in the literature (e.g. ℳ(10,3) →ℳ(8,3) studied by Klebanov et al <cit.>). With a slight twist, the renormalization group flows between multi-critical Lee-Yang fixed points ℳ(2,q) studied in the literature <cit.> fall into our proposal.  We claim that our flows give a complete understanding of the renormalization group flows between minimal models that preserve a modular tensor category (or more precisely modular fusion category) with the SU(2)_q-2 fusion ring.[The fusion ring SU(2)_q-2 is given by the fusion rule of the primary operators in the SU(2) WZW model at level q-2. It is essentially angular momentum addition of J = 0, 1/2, ⋯ ,q-2/2 with a q dependent “cap": J_1+J_2 + J_3 ≤ q-2. Given the fusion ring, the modular tensor category is further classified by the solutions of the Pentagon identity. The relation between the solutions of the Pentagon identity and the renormalization group invariants we will discuss can be found in <cit.>.]  

In our discussions, the non-invertible symmetries or categorical symmetries,[Recent applications of non-invertible symmetries in two-dimensional conformal field theories include <cit.>. ] which are realized by topological defect lines in two-dimensional conformal field theories, will play an analog of the 't Hooft anomaly. Similarly to the 't Hooft anomaly, their properties are preserved under the renormalization group flows. When the renormalization group invariants obtained from the non-invertible symmetries are different, they cannot be connected by the renormalization group flow that preserves the symmetries. We will show that our proposed renormalization group flows are not only consistent with the constraint but also give the actual flows between two minimal models with the same preserved non-invertible symmetries.

Some of our results may appeal to more physical intuitions. For instance, the preserved non-invertible symmetries in ℳ(p,4) →ℳ(p',4)  flows are given by the ℤ_2 Tambara-Yamagami modular tensor category in math terms or the duality defect line in physics terms. We know from mathematics that there are two distinct quantum dimensions consistent with the ℤ_2 Tambara-Yamagami (or Ising) fusion rule. We also know from physics that the quantum dimensions are renormalization group invariants. Accordingly, there exist two intrinsically different duality defects in two-dimensional conformal field theories. Our new renormalization group flows know them and the flows are completely separated as long as the duality is preserved. One physical application of such a flow was to identify the fate of the non-supersymmetric Yukawa fixed point in two dimensions <cit.>. As a vast generalization, our new results should give us a new map to explore renormalization group flows in two-dimensional quantum field theories.


The organization of our paper is as follows. In section <ref>, we give a review of Virasoro minimal models and topological defect lines. In section <ref>, we present our main claim of the new renormalization group flows and give supporting evidence from the study of the renormalization group invariants associated with the non-invertible symmetries. In section <ref>, we study several physically interesting examples of our new renormalization group flows. In section <ref>, we conclude with some discussions for future directions.









§ VIRASORO MINIMAL MODELS

Let us first state our conventions of Virasoro minimal models. We specify the Virasoro minimal model ℳ(p, q) by two coprime integers p and q. While our convention is more or less the same as the one in the yellow book <cit.>, one notable exception is they always assume p > q in the yellow book, but we take q to be a fixed integer, and we investigate the renormalization group flow that changes p. For instance, the well-known renormalization group flow from the tricritical Ising model to the critical Ising model is ℳ(5,4) →ℳ(3,4) in our paper rather than ℳ(5,4) →ℳ(4,3) as in the yellow book.





The central charge of ℳ(p, q) is 

    c = c̅ = 1 - 6(p - q)^2pq

and it has (p - 1)(q - 1)/2 (chiral) primary operators ϕ_(r, s) where r and s are the Kac indices whose ranges are 1 ≤ r < q, 1 ≤ s < p. Two primary operators ϕ_(r, s) and ϕ_(q -  r, p - s) are identified: they have the same conformal weight 

    h_r, s = h_q - r, p - s = (pr - qs)^2 - (p - q)^24pq.

The fusion rule of the primary operators is given by

    ϕ_(r, s)×ϕ_(m, n) = ∑_k = 1 + | r - m |
     k + r + m = 12^min (r + m - 1, 2q - 1 - r - m)∑_l = 1 + | s - n |
     l + s + n = 12^min (s + n - 1, 2p - 1 - s - n)ϕ_(k, l).

For example, we can choose a fundamental domain Γ of the Kac table, which specifies (p - 1)(q - 1)/2 distinct Kac indices as

    Γ = { (r, s) | 1 ≤ r ≤ q - 1, 1 ≤ s ≤ p - 1, pr + qs < pq }.

See Figure <ref>.




The Virasoro minimal models are further classified by the modular invariant partition functions. In this paper, we focus on those with A-series modular invariant partition functions (a.k.a. A-series minimal models). Since most of the renormalization group flows that we will discuss preserve the ℤ_2 symmetry, which can be used to relate the A-series and D-series by orbifolding (when it can be gauged), most of the following discussions apply to the D-series minimal models. Exceptions are half-integer k flow discussed in section <ref> because the deformation may not exist in D-series. Furthermore, when we can gauge the ℤ_2 symmetry, we have fermionic minimal models <cit.> as well. Our discussions also apply to their cases. Again, exceptions are half-integer k flow discussed in section <ref>. Our discussions do not directly apply to the E-series minimal models, however, and we need separate considerations. Some studies of the topological defect lines in E-series minimal models in view of the renormalization group flows can be found in <cit.>.







 §.§ Non-invertible symmetries and topological defect lines

In two-dimensional quantum field theories, non-invertible symmetries are synonyms of topological defect lines. In Virasoro minimal models with A-series modular invariant partition functions, it is widely believed that all the topological defect lines are given by the Verlinde lines.[To avoid trivial counterexamples, we here exclude space-time symmetries such as Poincaré transformation or space-time parity.] Verlinde lines L_(r,s) have the same label (r,s) as the (chiral) primary operators ϕ_(r, s) of the same theory, so  (p-1)(q-1)/2 of them are independent.

 The action of L_(r, s) on a primary state |ϕ_(ρ, σ)⟩ = ϕ_(ρ, σ)|0⟩ is given by

    L_(r, s)|ϕ_(ρ, σ)⟩ = S_(r, s), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, σ)⟩,

where S_(r, s), (ρ, σ) is modular S-matrix.  The explicit expression of the S-matrix is

    S_(r, s), (ρ, σ) = 2√(2pq)(-1)^1 + sρ + rσsin(πpqrρ)sin(πqpsσ),

which is real and symmetric.

As a special case, the action of L_(r, s) on the vacuum |0⟩ = |ϕ_(1, 1)⟩ is given by

    L_(r, s)|0⟩ = d_(r, s)|0⟩ = S_(r, s), (1, 1)S_(1, 1), (1, 1)|0⟩,

where the eigenvalue d_(r, s) is called quantum dimension of L_(r, s). The salient property of the quantum dimension is that it is a renormalization group invariant when L_(r,s) commutes with the deforming operators as we will see.

The Verlinde lines satisfy the same fusion rule as (chiral) primary operators:

    L_(r, s)× L_(m, n)   =  ∑ N^(k,l)_(r,s)(m,n) L_(k,l)   = ∑_k = 1 + | r - m |
     k + r + m = 12^min (r + m - 1, 2q - 1 - r - m)∑_l = 1 + | s - n |
     l + s + n = 12^min (s + n - 1, 2p - 1 - s - n) L_(k, l).

The fusion coefficients N^(k,l)_(r,s)(m,n) are related to the modular S-matrix by the Verlinde formula:

    N_ab^c = ∑_d S_ad S_bd S_dc/S_0d,

where we collectively denote (r,s) by a etc. This short-hand collective notation is sometimes used hereafter.



 §.§ Invertible TEZT symmetry

All the Virasoro minimal models with A-series modular invariance (except when p or q is 2) possess a special topological defect line L_(q - 1, 1) corresponding to the invertible ℤ_2 symmetry. It is invertible because the fusion rule (see e.g. (<ref>)) gives L_(q - 1, 1)× L_(q - 1, 1) = 1. 

 Under the action of L_(q - 1, 1), ℤ_2 charge of the primary operators ϕ_(r, s) can be summarized as  

    r-1      2   (when q is even) 
    
        s-1      2    (when p is even) 
    
        r+s-2      2    (when q and p are both odd)  .

See e.g. <cit.>.
The ℤ_2 symmetry is compatible with the fusion rule of the primary operators and operator identifications. If we gauge the ℤ_2 symmetry (or orbifold it in the conformal field theory language), A-series minimal models are exchanged with the D-series minimal models unless it is anomalous.

At this point, we note that the ℤ_2 symmetry is anomalous when p and q are both odd. One way to see this is to compute the quantum dimensions of L_(q - 1, 1), which is -1 rather than +1.  As we will review in the next subsection, the quantum dimension is a renormalization group invariant and it gives a selection rule of the renormalization group flow as the 't Hooft anomaly. We can also compute the spin contents of the defect Hilbert space, which will be again reviewed in the next subsection, and see that it is ±1/4 rather than 0,±1/2. As discussed in <cit.> this is a clear signal of the 't Hooft anomaly. Let us also recall that there is no D-series modular invariant partition function when p and q are both odd. This means that we cannot gauge the ℤ_2 invertible symmetry and it is consistent with the existence of the anomaly. 

In this paper, we focus on the renormalization group flow that preserves the ℤ_2 invertible symmetry. To be more precise, our renormalization group flow will preserve a larger modular tensor category with the SU(2)_q-2 fusion ring. This modular tensor category includes the ℤ_2 invertible symmetry but gives extra q-3 non-invertible symmetries.  In the next subsection, we will review how the preserved non-invertible symmetries will constrain the renormalization group flows.




 §.§ Renormalization group invariants

The salient feature of the topological defect lines regarded as non-invertible symmetries is that they give renormalization group invariants when they are preserved along the flow. Let us first discuss the condition that the topological defect lines are preserved along the renormalization group flow in the general setup.

Suppose we are at the ultraviolet conformal field theory with a topological defect line L_a. We deform the theory by a relevant operator ϕ_b. More generally we have to consider a collection of relevant operators {ϕ_b } to reach the desired infrared fixed point. In such a case, we assume that the following conditions are met with all the operators we need. Now, we say that L_a is preserved along the renormalization group flow when they commute with ϕ_b:  L_aϕ_b = ϕ_b L_a on the cylinder. More explicitly, they must satisfy

    L_a ϕ_b |Φ⟩ = ϕ_b L_a |Φ⟩

on any state |Φ⟩ on the cylinder. In unitary conformal field theories, it is sufficient to check it on the vacuum |0⟩ = |ϕ_(1, 1)⟩ <cit.>, but in non-unitary conformal field theories, we may have to study all the states. We will, however, explicitly see that in our discussions we do not gain any new constraint by studying all the states rather than the vacuum alone.

In the mathematics language, topological defect lines give a realization of the modular tensor category. As a modular tensor category, the physical properties of the topological defect lines are severely constrained. It is believed that the finite-rank modular tensor categories have no continuous parameters (so-called rigidity). Therefore, when the topological defect lines are preserved, we expect that the properties of the topological defect lines as a modular tensor category should be preserved by regarding the renormalization group flow as a continuous deformation.

Some properties of the topological defect lines are directly related to the 't Hooft anomaly i.e. an obstruction to gauge the symmetry, in which case it is not necessary to assume the non-existence of the continuous deformation of the modular tensor category to deduce the renormalization group invariance of the modular tensor category. It is simply the 't Hooft anomaly matching. 

In this paper, we focus on two such properties. One is the quantum dimensions of topological defect lines and the other is the spin contents of the defect Hilbert space. See e.g. <cit.> for studies of these properties in various examples.

We have already introduced the quantum dimensions of the topological defect lines L_a in (<ref>). 
 The renormalization group invariance of the quantum dimensions can be understood as follows. Recall that the fusion of the topological defect lines is

    L_a × L_b = ∑_c N_ab^cL_c,

where N_ab^c is the fusion coefficients. 

Let us now take the vacuum expectation value of this equality: ⟨ L_a ⟩ := ⟨ 0|L_a | 0 ⟩, where |0⟩ = |ϕ_(1,1)⟩ on the cylinder. The topological defect lines are topological, and we can place them far away. Then we may assume that the expectation value is factorized[Potentially, this argument can fail because we are applying the idea developed in unitary quantum field theories to non-unitary conformal field theories. It, however, turns out that the quantum dimensions that we can compute explicitly in non-unitary minimal models do satisfy the constraints obtained from this cluster decomposition ansatz.]

    ⟨ L_a ⟩×⟨ L_b ⟩ = ∑_c N_ab^c⟨ L_c ⟩.

The solutions of these quadratic equations turn out to be discrete, so it is invariant under the continuous renormalization group flows.

As the simplest example, let us consider the ℤ_2 invertible symmetry with the fusion rule L_(q - 1, 1)× L_(q - 1, 1) = L_(1, 1) (= 1). 
 By taking the vacuum expectation value, we can immediately see that ⟨ L_(q - 1, 1)⟩ = ± 1. The sign choice corresponds to the anomaly of the ℤ_2 invertible symmetry.[Whether one can choose the sign depends on the constraint from the other fusion rules. In Appendix C of <cit.>, we can find the discussion of why the minus sign is still consistent with the ℤ_2 selection rule in correlation functions.] In this case, the renormalization group invariance of the quantum dimension is equivalent to the 't Hooft anomaly matching. 


The other renormalization group invariants we will discuss are the spin contents of the defect Hilbert space associated with the preserved topological defect line L_a. To introduce the notion of the defect Hilbert space and the spin contents, we begin with the torus partition function with the insertion of L_a in the spatial direction.

    Z_L_a(τ, τ̅)    = Tr( L_a q^L_0-c/24q̅^L̅_0-c/24)    = ∑_b S_a b/S_0bχ_b(τ) χ̅_b(τ̅) ,

where q= e^2π i τ and χ_b is the Virasoro character of the primary operator ϕ_b.

Let us perform the S-modular transformation (τ̃ = -τ^-1) to regard the partition function as the sum over the defect Hilbert space ℋ_L_a:

    Z_L_a(τ, τ̅)    = ∑_b, c, dS_a b/S_0b S_bc S_bdχ_c(τ̃) χ̅_d(τ̅̃̅)     = ∑_c, d N_cd^aχ_c(τ̃) χ̅_d(τ̅̃̅)    = Tr_ℋ_L_a(  q̃^L_0-c/24q̅̃̅^L̅_0-c/24),

where we have used the Verlinde formula (<ref>). In the last line, the topological defect line is inserted in the time direction, and the quantization condition on the spatial circle is modified.
The spin contents of the defect Hilbert space ℋ_L_a are defined by the set of “Lorents spin" { h_c - h_d  (mod ℤ) } evaluated over non-zero N_cd^a. 
 

Let us now argue that the spin contents of preserved topological defect lines are renormalization group invariants (see e.g. <cit.>). The idea is that the topological defect lines are topological and the relevant deformation preserves the U(1) rotational symmetry of the quantum field theory. It implies that the twisted quantization condition on the defect Hilbert space cannot change in a discrete manner. More precisely, in <cit.> it is claimed that the spin content of the infrared theory must be a subset of the ultraviolet theory. The spin contents of the infrared theory can be a subset because the states might become heavy and decouple.





§ NEW RENORMALIZATION GROUP FLOWS

In this section, based on the study of non-invertible symmetries, we propose infinitely many new renormalization group flows in Virasoro minimal models. We give some formal checks of the agreement of renormalization group invariants under the flow. Examples will be given in the next section.



 §.§ Renormalization group flow TEXT


The main claim of our paper is there exist infinitely many renormalization group flows ℳ(kq + I, q) →ℳ(kq-I, q) induced by the ϕ_(1, 2k + 1) deformation, which satisfy the constraint from the non-invertible symmetries. We further claim that the renormalization group flow is complete when we preserve a modular tensor category with the  SU(2)_q-2 fusion rule that includes the ℤ_2 invertible symmetry. By complete, we mean that all the possible flows that preserve a modular tensor category with the SU(2)_q-2 fusion ring can be represented by a repeated use of our proposed flows ℳ(kq + I, q) →ℳ(kq-I, q).
For concreteness, we assume k and I are both integers in this subsection. The generalization for non-integer k (and I) will be discussed in the next subsection. The half-integer case does not preserve the modular tensor category with the SU(2)_q-2 fusion ring.

We first show that the ϕ_(1, 2k + 1) deformation preserves the SU(2)_q-2 subcategory L_(r, 1) (i= 1, ⋯, q-1) out of the non-invertible symmetries of the undeformed theory ℳ(kq + I, q). For this purpose, as reviewed in section <ref>, we should check L_(r,1) and ϕ_(1,2k+1) commute on the cylinder.

Let us start with the fusion rule (<ref>). We have

    ϕ_(1, 2k + 1)×ϕ_(ρ, σ) = ∑_l = 1 + | 2k + 1 - σ|
     l + σ = 02^min (2k + σ, 2(kq + I) - 2- 2k - σ)ϕ_(ρ, l) = ∑_l ∈Λϕ_(ρ, l)

in ℳ(kq + I, q), where we have defined Λ as 

    Λ = { l | 1 ≤ l ≤min (2k + σ, 2(kq + I) - 2- 2k - σ), l + σ = 0   mod 2  },

and then we can compare the action of L_(r, 1)ϕ_(1, 2k + 1) and ϕ_(1, 2k + 1) L_(r, 1)  on arbitrary primary states |ϕ_(ρ, σ)⟩:

    L_(r, 1)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩   = L_(r, 1)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩ = ∑_l ∈Λ L_(r, 1)ϕ_(ρ, l)|0⟩
       = ∑_l ∈Λ L_(r, 1)|ϕ_(ρ, l)⟩ = ∑_l ∈ΛS_(r, 1), (ρ, l)S_(1, 1), (ρ, l)|ϕ_(ρ, l)⟩
    ϕ_(1, 2k + 1)L_(r, 1)|ϕ_(ρ, σ)⟩   = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩ = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩
       = ∑_l ∈ΛS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(ρ, l)|0⟩ = ∑_l ∈ΛS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, l)⟩.

Because l + σ is even in the above sum,

    .S_(r, 1), (ρ, l)S_(1, 1), (ρ, l)/ .S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)   = .(-1)^(r - 1)lsin(πkp + I/prρ)sin(πkp + I/pρ)/ .(-1)^(r - 1)σsin(πkp + I/prρ)sin(πkp + I/pρ)
       = 1
 
and hence L_(1, 1), L_(2, 1), ⋯, L_(q - 2, 1), L_(q - 1, 1) commute with ϕ_(1,2k+1). This means that SU(2)_q-2 subcategory generated by L_(r,1) is preserved under the renormalization group flow induced by the ϕ_(1, 2k + 1) deformation. In particular, we stress that L_(q - 1, 1) generates the invertible ℤ_2 symmetry of the minimal models, so the renormalization group flow is ℤ_2 symmetric. 

In the above discussions, we realize that ϕ_(1,2k+1) can be replaced with ϕ_(1,2l+1) for any integer l such that  ϕ_(1,2l+1) is in the Kac table. The physical significance of choosing l=k is that then ϕ_(1,2k+1) will be the least relevant deformation within {ϕ_(1,2l+1)} if I<k.  When k>1, we typically have to introduce the other more relevant deformation ϕ_(1,2l+1) (with l<k) and fine-tune the deformation parameters to get to the critical point.[Generally we have to fine-tune k-1 parameters.] When I>k, we need fewer fine-tunings to reach the fixed point dictated by the proposed flow.



Let us now show that the quantum dimensions of all the preserved topological defect lines match under the proposed renormalization group flows ℳ(kq + I, q) →ℳ(kq-I, q).[The matching of the quantum dimensions when k=1 and k=2, corresponding to the previously known renormalization group flows, was discussed in <cit.>.] The proof follows by the direct calculation.
The quantum dimension d_(r, 1) of L_(r, 1) in ℳ(kq + I, q) is (- 1)^r + 1sin(πkq + I/qr)/sin(πkq + I/q) and that in ℳ(kq - I, q) is (- 1)^r + 1sin(πkq - I/qr)/sin(πkq - I/q). Its ratio is 

    sin(πkq + I/qr)sin(πkq - I/qr)sin(πkq - I/q)sin(πkq + I/q) = sin(πI/qr)sin(- πI/qr)sin(- πI/q)sin(πI/q) = 1.


We can further argue that the constraint from the quantum dimensions is sufficiently non-trivial. We observe that because kq + (± I + q) = (k + 1)q ± I, I takes a value in 1 ≤ I ≤ q - 1, and kq ± I and q are coprime, our proposed flows can be classified by φ(q) kinds of I, where φ(q) is Euler function. Now we want to see that each connected flows have different quantum dimensions. For this purpose, let us examine the quantum dimensions of L_(2,1) in ℳ(p,q) in particular; the constraint from the other preserved topological defect lines is typically not as strong. The explicit evaluation of the formula (<ref>) gives d_(2,1) = -2 cos(p/qπ) and we immediately observe d_(2,1) takes a distinct value for different p (mod q), given q. Thus, the quantum dimensions do distinguish our renormalization group flows.

From the general argument in section <ref>, under the new renormalization group flow, the spin contents of the preserved line  L_(r,1) (r = 1, ⋯, q - 1) should remain the same or become a subset.
While it may be non-trivial to check the agreement of all the spin contents in the most generic cases, one can systematically check the agreement in the following way. 

We first realize that N_(s, m)(t, n)^(r, 1)χ_(s, m)χ̅_(t, n) is non-zero only when m = n. Furthermore, the fusion coefficients necessary for us do not depend on m, so we can first determine the non-zero entry of N_(s, m)(t, n)^(r, 1)= δ_mn N_(s, 1)(t, 1)^(r, 1) from SU(2)_q - 2 fusion rule, and then we can compute the spin of the defect Hilbert space from (<ref>). 
In this way, it is possible to systematically compute the spin contents of L_(r,1) and see the agreement under the proposed renormalization group flow. We will show some examples in section <ref>.








Now let us show that the spin contents of ℋ_L_(2,1) in  ℳ(p,q) are preserved 
 under the proposed renormalization group flows. We first observe that the only non-zero fusion coefficients are N_(t, n) (t + 1, m)^(2, 1) and N_(t + 1, m) (t, m)^(2, 1). By computing ± (h_(t + 1, m) - h_(t, m)), we find the spin contents of ℋ_L_(2,1) are ±p/q1 + 2t/4 (mod 1/2ℤ) with t = 1, ⋯, q - 2. Thus, the spin contents of ℋ_L_(2,1) in ℳ(kq + I, q) contain those in ℳ(kq - I, q).

We further want to see that the spin contents of ℋ_L_(2,1) distinguish the renormalization group flows as the quantum dimensions. 
We observe that p/q1+2t/4 never equals to p/q1/4 mod ℤ/2, so p/q1/4  is always missing in the spin content. (Indeed, if this were true, we would have p/ql = ℤ, but since p and q are coprime, it should be a contradiction.) On the other hand, the missing ones (except 1/4q and 1/2q) always appear in p=2q-1 or p=2q-2 as long as q>3, so the spin contents of p=2q-1 and p=2q-2 are different from the other p's (mode q). We can repeat the analysis with the other p=2q-s with s=3, 4, ⋯, q-1 to realize that they all have different spin contents. The exceptional but easier q = 3 case can be treated separately (see section <ref>).

The renormalization group invariants cannot tell which of ℳ(kq+I,q) or  ℳ(kq-I,q) is the ultraviolet theory. To address this question, let us mention two evidence that  ℳ(kq+I,q) is the ultraviolet theory. The first evidence is the c_eff theorem <cit.>. While ℳ(p,q) can be non-unitary, it is believed that as long as the deformation preserves the 𝒫𝒯-symmetry (see e.g. <cit.> for a review), the renormalization group flow shows the monotonicity with respect to the effective central charge c_eff = 1 -6/pq along the flow. If we accept the c_eff theorem (with the assumption our flows preserve the 𝒫𝒯-symmetry), we may conclude ℳ(kq+I,q) is the ultraviolet theory.[One caveat here is that we have not verified our renormalization group flow preserves the 𝒫𝒯 symmetry. It is an interesting question to see if the 𝒫𝒯 symmetry can be realized as a topological defect line.] 

The other evidence comes from the number of (singlet) relevant deformations <cit.>. It was argued there that along the renormalization group flow, the number of (singlet) relevant deformations should decrease.
Intuitively, if we deform the theory by relevant perturbation, the deformed theory becomes less critical and we expect less fine-tuning is needed to reach the criticality. In <cit.>, it was put on firmer ground by studying the topology of the renormalization group flow. Again once we accept this conjecture, we realize ℳ(kq+I,q) is the ultraviolet theory because it has a larger number of relevant singlet operators. Note also that if our constraint from the non-invertible symmetries did not exist, we could have found counterexamples of the conjecture. For instance in the would-be flow ℳ(7,4) →ℳ(5,4), the number of relevant singlet operators did not decrease. We have, however, forbidden the flow by matching the non-invertible symmetries. In this discussion, it is important to note that by singlet, we mean the operators that commute with all the non-invertible symmetries {L_(r,1)} whose fusion ring is SU(2)_q-2 than just those that commute with the ℤ_2 invertible symmetry.







Here are some historical comments. In a classic paper <cit.>, Zamolodchikov studied the renormalization group flow k=I=1, which turns out to be integrable <cit.>. This case is most physically interesting because it describes the flow between unitary minimal models. Subsequently, Ahn <cit.> and Lässig <cit.> proposed the generalization to the non-unitary minimal models independently, corresponding to k=1, I>1. Later, Dorey, Dunning and Tateo <cit.> (see also <cit.>) studied the case with k=2 from the viewpoint of non-linear integral equations. Our approach, based on the non-invertible symmetries, also provides us with new insights into the previously known flows.





 §.§ Half-integer TEXT

Our proposed renormalization group flow
 ℳ(kq + I, q) →ℳ(kq - I, q) induced by ϕ_(1, 2k + 1)
 formally makes sense even if k is a half-integer, where I takes a half-integer or integer depending on q. The main difference is that 2k+1 is even so not all the topological defect lines L_(r, 1) commute with ϕ_(1, 2k + 1), but only the subcategory spanned by L_(2ℓ + 1, 1), where ℓ is an integer, is preserved. This restriction can be explicitly seen in (<ref>), where for half-integer k, l+σ is odd (rather than even for integer k case discussed there), so we have to assume r is an odd integer. This subcategory has the fusion ring of SO(3)_[q/2]-1.
 
If we restrict ourselves to these topological defect lines, all the discussions above can be repeated. In particular, the consistency of the proposed renormalization group flow with the preserved non-invertible symmetry holds. For instance, the invariance of the  quantum dimensions of L_(r,1) can be checked as in (<ref>), but with half-integer k, the ratio becomes (-1)^r+1. Thus only the preserved lines with r=2ℓ+1 have the same quantum dimensions as claimed. The strongest constraint from the spin contents comes from L_(3,1) (rather than L_(2,1) which does not exist). In ℳ(p,q), the spin contents of ℋ_L_(3,1) are {0,±p/q(ℓ+1) } (ℓ=1, ⋯, q-3), and we can check it is consistent with our flow ℳ(kq + I, q) →ℳ(kq - I, q) with half-integer k.


Some specific renormalization group flows have been studied in the literature.
The particular case with k=1/2 was discussed in Dorey et al <cit.>, but note (p,q) there was swapped compared with our unified notation so that their ϕ_(1,2) is our ϕ_(2,1). The non-invertible symmetries, in this previously known case, were studied in <cit.>. Again, we generalize their works by letting k be an arbitrary half-integer.



Let us comment on the fate of the ℤ_2  invertible symmetry under the half-integer k flow.
 When k is a half-integer and q is odd, ϕ_(1, 2k + 1) is odd under the ℤ_2 invertible symmetry, so the preserved modular tensor category does not include L_(q - 1, 1). Nevertheless, after the flow, we do recover the entire modular tensor category with the SU(2)_q-2 fusion ring, including the invertible ℤ_2 symmetry. They must be emergent symmetries. While this must be the case since this half-integer k flow allows us to change the (non-)anomalous nature of the ℤ_2 symmetry, the microscopic details of the renormalization group flow may be of interest and needs further study.


 




§ SOME EXAMPLES


In this section, we study explicit examples of ℳ(kq+I,q) →ℳ(kq-I,q) flows for q=3,4,5, emphasizing physical interpretations. As a potentially exceptional case, we then examine q=2 separately.






 §.§ TEXT and anomaly matching


In this case, our proposal with integer k boils down to the conclusion that there exists distinct renormalization group flows among  ℳ(2l+1,3) and among ℳ(2l,3). The only non-trivial topological defect line preserved by the ϕ_(1, 2k + 1) deformation is L_(2,1) that generates the invertible ℤ_2 symmetry. The constraint from the quantum dimensions as well as spin contents will be equivalent to the 't Hooft anomaly matching. We recall that the ℤ_2 invertible symmetry is anomalous in  ℳ(2l+1,3) and non-anomalous in ℳ(2l,3) so they cannot mix under the ℤ_2 preserving renormalization group flow.

While the quantum dimensions of L_(2,1) can be directly computable from (<ref>), the spin contents of ℋ_L_(2,1) can be computed by the algorithm mentioned in section <ref>.
As argued there, non-zero entry of N_(s, m)(t, n)^(2, 1) implies m = n and (s, t) = (2, 1) or (1, 2), which have the opposite spins. From the dimensions of primary operators (<ref>), we find the spin of the defect Hilbert space ℋ_L_(2,1) is given by  ±(h_(2, m)-h_(1, m)) = ±1/4(p - 2m) with 1≤ m ≤ p - 1. The spin contents are therefore {±1/4} when p is odd and {0, ±1/2} 
 when p is even. 

We have summarized the quantum dimensions and spin contents of ℳ(p,3) models in Table <ref> and Table <ref>. Note that for the anomalous ℤ_2 invertible symmetry, the quantum dimension is -1 and the spin content is {±1/4}, which is consistent with the general argument <cit.>.








We would like to note that in the literature <cit.>, one of the flows in this class ℳ(10,3) →ℳ(8,3) induced by ϕ_(1,7) (k=3, I=1) was used to study the Landau-Ginzburg-like Lagrangian constructions of non-unitary minimal models. The further flow ℳ(8,3) →ℳ(4,3) induced by ϕ_(1,5) (k=2, I=2) can be straightforwardly realized in their Landau-Ginzburg-like Lagrangian. Another flow ℳ(7,3) →ℳ(5,3) induced by ϕ_(1, 5) (k = 2, I=1) was studied 
in <cit.> from the truncated conformal space approach.

Let us briefly discuss the case with half-integer k. With the half-integer k, ϕ_(1,2k+1) deformation does not preserve any non-trivial topological defect lines in ℳ(p,3), so the constraint is none. We, however, observe that some of the predicted flow such as ℳ(10,3) ϕ_(1, 6)⟶ℳ(5,3) ϕ_(1, 4)⟶ℳ (4,3) may give an interesting Lagrangian description of ℳ(5,3), given the Lagrangian description of ℳ(10,3) is available. Indeed, the flow has some theoretical interest because of the (dis)appearance of the anomalous ℤ_2 symmetry of ℳ(5,3) 
 in the middle of the renormalization group flow.








 §.§ TEXT and duality defects


Here, under the ϕ_(1, 2k + 1) deformations with integer k, the preserved topological defect lines are L_(1,1), L_(2,1) (sometimes called 𝒩) and L_(3,1) (sometimes called η) and generate the ℤ_2 Tambara-Yamagami modular tensor category. 
There are φ(4) = 2 distinct renormalization group flows predicted from our proposal. 


Let us first study the quantum dimensions of L_(2, 1) and L_(3, 1), as shown in Table <ref>. As discussed in section <ref>, d_(2,1) distinguishes the two renormalization group flows inside ℳ(p,4). Since the ℤ_2 invertible symmetry is non-anomalous, d_(3,1) =1 for all p, and it is not useful as a constraint on the renormalization group flow.



We can also compute the spin contents of L_(2,1) and L_(3,1). Since the topological defect line L_(3,1) generates the non-anomalous ℤ_2 invertible symmetry, the spin contents must be {0, ±1/2} as can be checked easily, so we focus on L_(2,1). The result of the explicit computation is summarized in Table <ref>.



As derived in section <ref>, for ℳ(p, 4), the general formula for h_s, m - h_t, n and the spin content of ℋ_L_(2,1) can be derived:

    h_s, m - h_t, n : {±3p - 8m16, ±5p - 8m16 | 1 ≤ m ≤ p - 1},
    the spin contents of ℋ_L_(2,1) : ±116, ±716    (p = 3, 5 mod 8)
    ±316, ±516    (p = 1, 7 mod 8)

This is compatible with (6.5) in <cit.> on the spin content of the duality defect line. With this agreement in mind, let us argue that L_(2,1) is indeed the duality defect line.

As is well-known in the critical Ising model ℳ(3,4) and the tricritical Ising model ℳ(5,4), the ℤ_2 Tambara-Yamagami modular tensor category has a physical interpretation as the duality defect line. We would like to offer a similar interpretation to the general ℳ(p,4) Virasoro minimal models.

First, let us review the duality defect in more generality <cit.>. Consider a quantum field theory 𝒬 on a d-dimensional spacetime with a non-anomalous ℤ_N q-form global symmetry. We divide the spacetime into two regions and gauge the ℤ_N global symmetry in half of the spacetime. Then, we can construct a codimension-one topological interface between the original theory 𝒬 and its gauged theory 𝒬/ℤ_N. If the original theory 𝒬 is invariant under gauging the ℤ_N global symmetry

    𝒬≅𝒬/ℤ_N ,

the interface becomes the duality defect in the original theory 𝒬. 

When do we expect the duality defect in this so-called half-space gauging construction? Note that the gauged theory has ℤ_N≅ℤ_N (d - q - 2)-form dual (or quantum) symmetry, so the ungauged theory and the gauged theory may have the same symmetry  as in (<ref>) if we require

    q = d - 22.

Our cases discussed in this paper correspond to  (d, q) = (2, 0).

Given a duality defect in the ultraviolet conformal field theory, we want to understand its fate under the renormalization group flow. Let us assume that the duality procedure commutes with the deformation of the ultraviolet conformal field theory.[This assumption is stronger than just assuming the deformation preserves ℤ_2 invertible symmetry. For the difference, see the example of half-integer k flows below.] Then, we should have the duality defect with the same property in the infrared as well. Such an existence should provide us with a strong constraint on the renormalization group flow.




We want to argue that the topological defect line L_(2,1) in ℳ(p,4) is the duality defect line. Let us consider general ℳ(p,q) with the A-series modular invariant partition functions Z_A_q - 1, A_p - 1. Let us further assume the case when p or q is even (but not 2) so that the ℤ_2 symmetry is non-anomalous and can be gauged.
The partition function of the ℤ_2 gauged theory can be obtained by the orbifold technique:

    Z_ℳ(p, q)/ℤ_2 = 
            Z_D_q/2 + 1, A_p - 1   when q = 0 mod 2
    
            Z_A_q - 1, D_p/2 + 1   when p = 0 mod 2.
 
The right-hand side is generically a non-diagonal modular invariant partition function of D-series, but the exception is when q=4 (or p=4). 
Because A_3 ≅ D_3, A-series minimal models ℳ(p, q) are invariant under gauging ℤ_2 global symmetry if and only if p or q is 4. In other words,  ℳ(4, q) is self-dual under the ℤ_2 gauging and it has a duality defect line 𝒩. Moreover, we can show that the duality defect line 𝒩 satisfies the ℤ_2 Tambara-Yamagami fusion rule 
 such as η×η = 1, η×𝒩 = 𝒩×η = 𝒩 and  𝒩×𝒩 = 1+η.
This is nothing but the fusion rule of the topological defect lines L_(2,1) and L_(3,1) preserved under the renormalization group flow ℳ(4k+I,4) →ℳ(4k-I,4) induced by ϕ_(1,2k+1). 


After seeing how the duality line plays a significant role in the renormalization group flows of  ℳ(4k+I,4) →ℳ(4k-I,4) when k is an integer, let us briefly discuss the case when k is a half-integer. Here, ϕ_(1,2k+1) does not commute with L_(2,1) and the only preserved symmetry is L_(3,1). Accordingly, there are no obstructions for the renormalization group flows between any pair of p in ℳ(p,4). The simplest example would be ℳ(3,4)
 to “ℳ(1,4)" induced by ϕ_(1,2) (= ϵ), which is a massive flow. Physically, if we change the temperature of the Ising model, the Kramers–Wannier duality is broken and the criticality will be lost. See <cit.> for an interpretation of “ℳ(1,4)" from the viewpoint of the integral equations.






To conclude this subsection, we have summarized the renormalization group flows between ℳ(p,4) in Figure <ref>. As a physical application beyond the classic tri-critical Ising to critical Ising flow ℳ(5,4) →ℳ(3,4), the fermionic version of the renormalization group flow ℳ(11,4) to ℳ(5,4) was discussed in <cit.> to understand the fate of the non-supersymmetric Yukawa fixed point.














 §.§ TEXT 


We repeat our analysis on the renormalization group flows ℳ(5k+I,5) →ℳ(5k-I,5) induced by ϕ_(1,2k+1) that preserve 
 a modular tensor category with the SU(2)_3 fusion ring, assuming k is an integer. Here, we have φ(5)=4 distinct flows that cannot mix. The 't Hooft anomaly matching of the ℤ_2 invertible symmetry distinguishes ℳ(p,5) with even p and odd p, but the other non-invertible symmetries give a finer classification.

The quantum dimensions of preserved non-invertible symmetries are given in Table <ref>. Here, ϕ = 1+√(5)/2 is the golden ratio. As discussed in section <ref>, not only all the quantum dimensions are consistent with the proposed renormalization group flow, but L_(2,1) gives the strongest constraint such that it fully distinguishes our proposed renormalization group flows.


Let us also compute the spin contents of the preserved topological defect lines. On  ℋ_L_(2,1), the non-zero fusion coefficients are N_(1, m)(2, m)^(2, 1), N_(2, m)(1, m)^(2, 1),  N_(2, m)(3, m)^(2, 1), N_(3, m)(2, m)^(2, 1), N_(3, m)(4, m)^(2, 1), and N_(4, m)(3, m)^(2, 1), so we have

    {±(3p/20 -m/2), ±1/4(p-2m), ±(7p/20 -m/2)  } ,

where 1 ≤ m ≤ p - 1. Similarly on ℋ_L_(3,1), from the non-zero fusion coefficients N_(1, m)(3, m)^(3, 1), N_(3, m)(1, m)^(3, 1), N_(2, m)(4, m)^(3, 1), N_(4, m)(2, m)^(3, 1), N_(2, m)(2, m)^(3, 1), and N_(3, m)(3, m)^(3, 1), we have

    {±(2p/5-m), ±(3p/5 -m), 0 } .

Finally, on ℋ_L_(4,1), from the non-zero fusion coefficients N_(1, m)(4, m)^(4, 1), N_(4, m)(1, m)^(4, 1), N_(2, m)(3, m)^(4, 1), and N_(3, m)(2, m)^(4, 1),  we have 

    {±3/4(p-2m), ±1/4(p-2m) } .



The results are summarized in Table <ref>. We note that the constraints from the quantum dimensions and constraints from the spin contents are the same.






Since this is the first example of the appearance of p=2 as in the flow ℳ(8,5) →ℳ(2,5) induced by ϕ_(1, 3), let us comment on it. We will give more detailed discussions in section <ref>. Unlike in ℳ(p,5) with p>2, in ℳ(2,5), L_(4,1) is identified with L_(1,1) and L_(3,1) is identified with L_(2,1), so we have only half of the preserved non-invertible symmetries in the infrared compared with the ones in the ultraviolet. These symmetries may be spontaneously broken or disappear. Note, in particular, that The ℤ_2 invertible symmetry generated by L_(4,1) is non-anomalous, so it can simply disappear. Understanding the fate of these symmetries requires dynamical analysis which is beyond the scope of our paper, but further discussions follow in section <ref>.


Let us finally make a brief discussion on the half-integer k flow. The preserved topological defect lines are L_(1,1) and L_(3,1), which form the fusion rule of ℳ(2,5) (Lee-Yang fusion ring). It is also called the Fibonacci fusion ring (see the appearance of the golden ratio above). There exist two distinct renormalization group flows depending on d_(3,1), which is consistent with the classification of the modular tensor category with the Fibonacci fusion ring.
The ℤ_2 invertible symmetry is broken by the deformation, but it reappears as an emergent symmetry. This is how we can change the (non-)anomalous nature of the ℤ_2 invertible symmetry along the renormalization group flow.
 



 §.§ Flows to TEXT

As far as the non-invertible symmetry goes, there is nothing wrong with our flows ℳ(kq + I, q) →ℳ(kq-I, q) even when q=2 or kq-I = 2. A closer look tells us, however, the situation is more subtle essentially because there is no ℤ_2 invertible symmetry in ℳ(p,q) when either p or q is 2. From the viewpoint of the topological defect lines, the would-be  ℤ_2 topological defect line L_(q - 1, 1) is identified with the identity defect line L_(1, 1). In other words, there exist no charged primary operators under the ℤ_2 invertible symmetry.

By taking q = 2, our proposal suggests the existence of the renormalization group flow ℳ(2k + I, 2) →ℳ(2k - I, 2) induced by ϕ_(1, 2k + 1). The case with I=1, however, may sound problematic because  ϕ_(1,2k + 1) is outside of the Kac table. The most reasonable interpretation is that the flow exists but the primary operator that induces the renormalization group flow is a certain fine-tuned combination of ϕ_(1, l) with l=1,2, ⋯, k.[Under the identification, this is equivalent to taking l=1,3, ⋯, 2k-1.] Indeed, the numerical analysis based on the truncated conformal space approach shows the existence of such flows (for small k) <cit.>. Of course, we could be simply agnostic about these flows because the argument based on the non-invertible symmetries does not give any constraint at all here, but we are tempted to unify them within our proposal.

Another subtle case is q≠ 2 but kq - I = 2. In this case, the question is about the fate of the ℤ_2 invertible symmetry.
We start with the ultraviolet theory with a ℤ_2 invertible symmetry and we add the deformation that preserves it. In the infrared limit, we end up with the theory that does not possess any ℤ_2 invertible symmetry. What happened to the symmetry? One possibility is it is spontaneously broken. Another possibility is all the charged operators are decoupled in the infrared limit. 
 The latter is possible because the ℤ_2 symmetry here is non-anomalous.
 
 In the massive flow example of ℳ(4,3) →ℳ(2,3) induced by ϕ_(1,3) (formally k=1, I=1 flow), where a Lagrangian description is available, both can happen depending on the sign of the deformation. Regarding ℳ(4,3) as a critical ϕ^4 theory, adding m^2ϕ^2 with positive m^2 gives decoupling of the ℤ_2 charged field ϕ, and with negative m^2 it gives spontaneous breaking of the ℤ_2 symmetry.

Physically ℳ(2, q) can be realized in the multi-critical Lee-Yang fixed point in two dimensions.
So far, the canonical construction of the (multi-critical) Lee-Yang fixed point was to start with the ℤ_2 symmetric theory, say Ising model, and add the (imaginary) ℤ_2 breaking interaction, say pure imaginary magnetic field, to get to the fixed point. This new construction without explicitly breaking the ℤ_2 symmetry is novel and worth studying further.




§ DISCUSSIONS


In this paper, based on the study of non-invertible symmetries, we have proposed there exist infinitely many new renormalization group flows between Virasoro minimal models  ℳ(kq + I, q) →ℳ(kq-I, q) induced by ϕ_(1,2k+1). They vastly generalize the previously proposed renormalization group flows in the literature.

One should be able to check our new renormalization group flows by using the conformal perturbation theory or truncated conformal space approach <cit.>. Here, let us simply observe that the deformation operator ϕ_(1,2k+1) is more relevant when the separation of the renormalization group flow given by  I is larger. This agrees with our intuitive picture; the more relevant deformations, the longer the renormalization group trajectory becomes. It should also agree with the difference of the central charge at the fixed points. More recent approaches to the renormalization group flow based on the boundary states and interfaces can be found in <cit.>, and they may be useful to understand our new renormalization group flows better.
 

For a future direction, it is interesting to study the integrable structure of our new renormalization group flows  ℳ(kq + I, q) →ℳ(kq-I, q). As initiated in <cit.> the flows with I=1 (or I=1/2) can be described by the Thermodynamic Bethe Ansatz <cit.>. When I>1, there seem to be no known descriptions in terms of the Thermodynamic Bethe Ansatz, but the integral equations that describe the effective central charge along the flow were proposed 
 <cit.>. It should be interesting to generalize these works to our new renormalization group flows.


Another direction to be explored further is a concrete realization of our new renormalization group flows in statistical models or quantum field theories. It is known that any (non-unitary) minimal models have a statistical model realization based on the Restricted Solid-On-Solid (RSOS) model <cit.> (see also <cit.> for the related spin chain construction). It is interesting to understand the nature of the anomalous ℤ_2 symmetries, duality defects as well as general non-invertible symmetries preserved by our renormalization group flow.

As for the quantum field theory realization, we are still in our infancy even more than fifty years after the birth of conformal field theories. This is because the quantum field theory descriptions (e.g. Landau-Ginzburg theory <cit.> or gauge theory <cit.>) for non-unitary minimal models are largely unknown. We, nevertheless, note that 2d-4d correspondence <cit.> gives realizations of certain (but probably not all) non-unitary chiral algebra from supersymmetric conformal field theories in four dimensions. 
We hope our renormalization group flow may become a breakthrough in this direction. 

Concerning the explicit constructions of minimal models from known quantum field theories, while our discussions focus on the infrared flow, it seems interesting to look at the flow in the opposite way as an ultraviolet flow <cit.>. Since the ultraviolet flow is a non-renormalizable deformation, we may need some other input such as an integrability to pursue, but once done, it should give a strong hint toward the quantum field theory descriptions of the ultraviolet theory whose “Lagrangian description" were unknown.

Why do we crave Lagrangian descriptions of the Virasoro minimal models?
We wish to point out here that once the Lagrangian descriptions are available, we may generalize these models in higher dimensions. When we obtain the higher dimensional fixed points, we can ask whether the constraints on the renormalization flow from the non-invertible symmetries can also be uplifted. Since we have less explicit constructions of non-invertible symmetries in higher dimensions, this is an outstanding question to be pursued.



§ ACKNOWLEDGMENTS

TT is supported by JSPS KAKENHI Grant Number JP24KJ1500.
YN is in part supported by JSPS KAKENHI Grant Number 21K03581. YN would like to thank C. Ahn and K. Kikuchi for the correspondence and discussions.


h-physrev






intersections, calc, arrows












When we insert L_(r, s) along time direction, the torus partition function which is defined by a trace over defect Hilbert space ℋ_L_(r, s) is written as

Consider the torus partition function. Let us further insert  L_(r, s) along the time direction. The torus partition function with the defect inserted can be interpreted as the trace over the defect Hilbert space ℋ_L_(r, s):

    Z_L_(r, s)(τ, τ̅) = ∑_(r^', s^'), (ρ^', σ^') ∈Γ n_(r^', s^'), (ρ^', σ^')χ_(r^', s^')(τ)χ̅_(ρ^', σ^')(τ̅),

where n_(r^', s^'), (ρ^', σ^') is non-negative integer, τ is modular parameter and χ_(r^', s^')(τ) is (degenerate) Virasoro character.   When n_(r^', s^'), (ρ^', σ^')≠ 0, the spin content of ℋ_L_(r, s) contains h_r^', s^' - h_ρ^', σ^' mod ℤ where h_r^', s^' is the conformal dimension of the primary operator ϕ_(r^', s^').

To compute Z_L_(r, s)(τ, τ̅), we use the modular S-transformation. Because of modular S invariance, 

    Z_L_(r, s)(- 1τ, - 1τ̅) = Z^L_(r, s)(τ, τ̅) := Tr_ℋ(L_(r, s)q^L_0 - c/24q̅^L̅_0 - c/24)

where in the right hand side, the topological defect lines are inserted in the space direction. Once we know the action of L_(r,s) on primary states, the right hand side is explicitly computable. 

From (<ref>) and (<ref>), we get

    𝖫 = 𝖲𝖭𝖲

where 𝖫 = (𝖫_IJ), 𝖲 = (𝖲_IJ) and 𝖭 = (𝖭_IJ) are matrices defined by

    𝖫_IJ = diag(S_(r, s), (1, 1)S_(1, 1), (1, 1), ⋯), 𝖲_IJ = S_IJ, 𝖭_IJ = n_IJ,

where I, J = (r^', s^'), (ρ^', σ^') are multi indices. 





Potentially interesting new examples.

ℳ(5,3) →ℳ(4,3), 

ℳ(10,3) →ℳ(5,3) (this can give an interesting “Lagrangian" description of ℳ(5,3).)

ℳ(7,4) →ℳ(5,4) (this preserves only ℤ_2), 

ℳ(8,5) →ℳ(7,5) 


 


§ NOTATION

First, we check the notation of minimal models in this paper. Minimal models denoted by ℳ(p, q) are specified by two coprime integers p and q. p > q in the yellow book, but in this paper, we fix q to a fixed integer and investigate the renormalization group flow among ℳ(p, q) series where q is fixed. So the critical Ising model is denoted by ℳ(3, 4) rather than ℳ(4, 3) in the yellow book when we study ℳ(p, 4) series.



The central charge of ℳ(p, q) is 

    c = c̅ = 1 - 6(p - q)^2pq

and it has (p - 1)(q - 1)/2 primary fields ϕ_(r, s) where p and q are the Kac indices and meets 1 ≤ r < q, 1 ≤ s < p. Two primayy fields ϕ_(r, s) and ϕ_(q -  r, p - s) are identified which has the same conformal weight 

    h_r, s = h_q - r, p - s = (pr - qs)^2 - (p - q)^24pq.

The fusion rule of the primary fields is given by

    ϕ_(r, s)×ϕ_(m, n) = ∑_k = 1 + | r - m |
     k + r + m = 12^min (r + m - 1, 2q - 1 - r - m)∑_l = 1 + | s - n |
     l + s + n = 12^min (s + n - 1, 2p - 1 - s - n)ϕ_(k, l).

We choose the fundamental domain in the Kac table composed of (p - 1)(q - 1)/2 distinct Kac indices as below:

    Γ = { (r, s) | 1 ≤ r ≤ q - 1, 1 ≤ s ≤ p - 1, ps + qr < pq }



§ RENORMALIZATION GROUP FLOW OF   MINIMAL MODELS



 §.§ Non-invertible symmetry

Consider a QFT 𝒬 on a d-dimensional spacetime with a non-anomalous ℤ_N q-form global symmetry. We divide the spacetime into two regions and gauge ℤ_N global symmetry in half of the spacetime. Then, we can construct a codimension-one topological interface between the original theory 𝒬 and its gauged theory 𝒬/ℤ_N. If the original theory 𝒬 is invariant under gauging ℤ_N global symmetry

    𝒬≅𝒬/ℤ_N ,

the interface becomes the duality defect in the original theory 𝒬. The gauged theory has ℤ_N≅ℤ_N (d - q - 2)-form dual(quantum) symmetry. (<ref>) can be the case where

    q = d - 22.

In particular, q = 0 if d = 2. The operation of discrete gauging is topological and commutes with renormalization group flow. So the existence of the duality defect is renormalization group invariant. It gives us the strong constraint on renormalization group flow.



In this paper, we consider A-series minimal models ℳ(p, q) in two-dimensional conformal field theory, whose partition functions are diagonal modular invariant. They have ℤ_2 0-form global symmetry. When p and q are both odd, this symmetry is anomalous. We denote the partition function of A-series minimal models ℳ(p, q)(p > q) as Z_A_q - 1, A_p - 1, then the partition function of its gauged theory is

    Z_ℳ(p, q)/ℤ_2 = 
            Z_D_q/2 + 1, A_p - 1   when q = 0 mod 2
    
            Z_A_q - 1, D_p/2 + 1   when p = 0 mod 2.

Because A_3 ≅ D_3, A-series minimal models ℳ(p, q) is invariant under gauging ℤ_2 global symmetry if and only if p or q is 4. We fix q to 4 below. From the above, ℳ(p, 4) must flow to ℳ(p^', 4).




 §.§  -theorem

Zamolodchikov showed that the central charge decreases along renormalization group flow between two unitary CFTs. For non-unitary case, there is generalized c-theorem which states the effective central charenormalization groupe c_eff = c - 24h_min do not increase along renormalization group flow which connects two 𝒫𝒯 invariant conformal field theories. For A-series minimal models ℳ(p, q), c_eff = 1 - 6/pq. Therefore ℳ(p, 4) must flow to ℳ(p^', 4) where p > p^'.




 §.§ Spin content

First, we review some properties of topological defect lines of A-series minimal models ℳ(p, q). ℳ(p, q) has (p - 1)(q - 1)/2 topological defect lines (Verlinde lines) L_(r, s) for each of the primary fields ϕ_(r, s). The action of L_(r, s) on a primary state |ρ, σ⟩ = ϕ_(ρ, σ)|0⟩ is given by

    L_(r, s)|ϕ_(ρ, σ)⟩ = S_(r, s), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, σ)⟩,

where S_(r, s), (ρ, σ) is modular S-matrix and given by

    S_(r, s), (ρ, σ) = 2√(2pq)(-1)^1 + sρ + rσsin(πpqrρ)sin(πqpsσ).

The action of L_(r, s) on the vacuum is given by

    L_(r, s)|0⟩ = d_(r, s)|0⟩ = S_(r, s), (1, 1)S_(1, 1), (1, 1)|0⟩,

where the eigenvalue d_(r, s) is called quantum dimension of L_(r, s). 




When we insert L_(r, s) along time direction, the torus partition function which is defined by a trace over defect Hilbert space ℋ_L_(r, s) is written as 

    Z_L_(r, s)(τ, τ̅) = ∑_(r^', s^'), (ρ^', σ^') ∈Γ n_(r^', s^'), (ρ^', σ^')χ_(r^', s^')(τ)χ̅_(ρ^', σ^')(τ̅),

where n_(r^', s^'), (ρ^', σ^') is non-negative integer, τ is modular parameter and χ_(r^', s^')(τ) is Virasoro character. The way to read the spin content of the defect Hilbert space ℋ_L_(r, s) is following. Because of modular S invariance, 

    Z_L_(r, s)(- 1τ, - 1τ̅) = Z^L_(r, s)(τ, τ̅) := Tr_ℋ(L_(r, s)q^L_0 - c/24q̅^L̅_0 - c/24)

From (<ref>) and (<ref>), we get

    𝖫 = 𝖲𝖭𝖲

where 𝖫 = (𝖫_IJ), 𝖲 = (𝖲_IJ) and 𝖭 = (𝖭_IJ) are matrices defined by

    𝖫_IJ = diag(S_rs, 11S_11, 11, ⋯), 𝖲_IJ = S_IJ, 𝖭_IJ = n_IJ,

where I, J = (r^', s^'), (ρ^', σ^') are multi indices. When 𝖭_IJ≠ 0, the spin content of ℋ_L_(r, s) contains h_I - h_J = h_r^', s^' - h_ρ^', σ^' mod ℤ where h_r^', s^' is the conformal dimension of the primary field ϕ_(r^', s^'). If L_(r, s) survives under renormalization group flow, the spin content of the defect Hilbert space ℋ_L_(r, s) in IR must be a subset of that in UV. 




We insert the duality defect line 𝒩 = L_(2, 1), generated by ℤ_2 0-form symmetry whose topological defect line is η = L_(3, 1), along time direction and compute the spin content of the defect Hilbert space ℋ_𝒩 for p = 3, 5, ⋯, 17. The result is in Table <ref>.



In general, we can expect that h_r^', s^' - h_ρ^', σ^' and the spin content of ℋ_𝒩 are

    h_r^', s^' - h_ρ^', σ^' : {±3p - 8i16, ±5p - 8i16 | 1 ≤ i ≤ m},
    the spin content of ℋ_𝒩 : ±116, ±716    (p = 3, 5 mod 8)
    ±316, ±516    (p = 1, 7 mod 8)

when we consider minimal model ℳ(p = 2m + 1, 4).
From the above, the following renormalization group flow can be inferred.


§ RG FLOW TEXT

Zamolodchikov states that there is a flow ℳ(q + 1, q) →ℳ(q -1, q) by ϕ_(1, 3) perturbation in and Ahn and Lässig state that there is a flow ℳ(q + I, q) →ℳ(q - I, q). Moreover, Dorey, Dunning and Tateo states that there is a flow ℳ(2q + I, q) →ℳ(2q - I, q). From these statements and the previous section, we can infer that there is a flow ℳ(kq + I, q) →ℳ(kq - I, q) by ϕ_(1, 2k + 1) perturbation. Because kq + (± I + q) = (k + 1)q ± I, I takes a value in 1 ≤ I ≤ q - 1. Moreover, kq ± I and q are coprime, so there are φ(p) kinds of I where φ(q) is Euler function. We show this conjectural flow below. Our strategy is to find the topological defect lines which commute with ϕ_(1, 2k + 1) and calculate these quantum dimensions which are renormalization group flow. This invariance can be understood as below. The fusion of the topological defect lines is

    L_i × L_j = ∑_k N_ij^kL_k

where N_ij^k is the fusion coefficients and take the vacuum expectaion value. The defect lines are topological, so we can place them far away and it is factorized

    ⟨ L_i ⟩×⟨ L_j ⟩ = ∑_k N_ij^k⟨ L_k ⟩.

The solution of these quadratic equations is discrete, hence it is invariant under continuous renormalization group flow.



From (<ref>), we get

    ϕ_(1, 2k + 1)×ϕ_(ρ, σ) = ∑_l = 1 + | 2k + 1 - σ|
     l + σ = 02^min (2k + σ, 2(kq + I) - 2- 2k - σ)ϕ_(ρ, l) =: ∑_some l'sϕ_(ρ, l).

in ℳ(kq + I, q) and then

    L_(r, 1)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩   = L_(r, 1)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩ = ∑_some l's L_(r, 1)ϕ_(ρ, l)|0⟩
       = ∑_some l's L_(r, 1)|ϕ_(ρ, l)⟩ = ∑_some l'sS_(r, 1), (ρ, l)S_(1, 1), (ρ, l)|ϕ_(ρ, l)⟩
    ϕ_(1, 2k + 1)L_(r, 1)|ϕ_(ρ, σ)⟩   = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩ = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩
       = ∑_some l'sS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(ρ, l)|0⟩ = ∑_some l'sS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, l)⟩.

Because l + σ is even in the above sum,

    .S_(r, 1), (ρ, l)S_(1, 1), (ρ, l)/ .S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)   = .(-1)^(r - 1)lsin(πkp + I/prρ)sin(πkp + I/pρ)/ .(-1)^(r - 1)σsin(πkp + I/prρ)sin(πkp + I/pρ)
       = 1

and hence L_(1, 1), L_(2, 1), ⋯, L_(q - 2, 1), L_(q - 1, 1) whose fusion is closed commute with ϕ_(1, 2k + 1) and they survive under renormalization group flow. The quantum dimension d_(r, 1) of L_(r, 1) in ℳ(kq + I, q) is (- 1)^r + 1sin(πkq + I/qr)/sin(πkq + I/q) and that in ℳ(kq - I, q) is (- 1)^r + 1sin(πkq - I/qr)/sin(πkq - I/q). Its ratio is 

    sin(πkq + I/qr)sin(πkq - I/qr)sin(πkq - I/q)sin(πkq + I/q) = sin(πI/qr)sin(- πI/qr)sin(- πI/q)sin(πI/q) = 1.



§ DISCUSSION



    
  * a puzzle when q=2. OK
    
  * Generalization of (2,1)  OK
    
  * deformations.Conformal perturbation theory.
    
  * conformal dimensions of ϕ_1,2k+1 and the length of RG flows.
    
  * RG flow from Cardy's ansatzs.
    
  * Gaiotto's RG defect




3/2= 3+0/2 = 2+1/2


2/2= 1+3/2 = 0+2/2 =1+1/2 = 2+2/2?


1/2= 1+0/2 = 3+2/2 = 1+2/2

 Paper Title: ISDE with logarithmic interaction and characteristic polynomials.



A Multi-Reference Relaxation Enforced Neighborhood Search Heuristic in SCIP
    
Suresh Bolusani^*10000-0002-5735-3443 Gioni Mexi10000-0003-0964-9802Mathieu Besançon20000-0002-6284-3033 Mark Turner10000-0001-7270-1496

    August 5, 2024
==============================================================================================================================================



§ ABSTRACT  We consider certain random matrix eigenvalue dynamics, akin to Dyson Brownian motion, introduced by Rider and Valko <cit.>. This is a multi-dimensional generalisation of a one-dimensional diffusion studied by Matsumoto and Yor <cit.>. We show that from every initial condition, including ones involving coinciding coordinates, the dynamics, enhanced with more information, converge on path-space to an infinite-dimensional Feller-continuous diffusion process. We show that the limiting diffusion solves an infinite-dimensional system of stochastic differential equations (ISDE) with logarithmic interaction. Moreover, we show convergence in the long-time limit of the infinite-dimensional dynamics starting from any initial condition to the equilibrium measure, given by the inverse points of the Bessel determinantal point process. As far as we can tell, this is: (a) the first path-space convergence result of random matrix dynamics starting from every initial condition to an infinite-dimensional Feller diffusion, (b) the first construction of solutions to an ISDE with logarithmic interaction from every initial condition for which the singular drift term can be defined at time 0, (c) the first convergence to equilibrium result from every initial condition for an ISDE of this kind. The argument splits into two parts. The first part builds on the method of intertwiners introduced and developed by Borodin and Olshanski <cit.>. The main new ingredients are a uniform, in a certain sense, approximation theorem of the spectrum of a family of random matrices indexed by an infinite-dimensional space and an extension of the method of intertwiners to deal with convergence to equilibrium. The second part introduces a new approach towards convergence of the singular drift term in the dynamics and for showing non-intersection of the limiting paths via certain “characteristic polynomials" associated to the process. We believe variations of it will be applicable to other infinite-dimensional dynamics coming from random matrices.







§ INTRODUCTION





 


 §.§ Background


The rigorous study of the non-equilibrium statistical mechanics of an infinite system of particles interacting via a potential was initiated in the seminal works <cit.> of Lanford. The study of the stochastic version of this problem, when one adds independent Brownian forces, and the dynamics are now described by an infinite-dimensional system of stochastic differential equations (henceforth abbreviated ISDE) was then first taken up by 
Lang <cit.> who proved well-posedness of the equilibrium dynamics for certain nice potentials. Many authors then contributed to this program, including Doss, Royer, Rost, Lippner and Fritz, and extended such results to non-equilibrium dynamics and more general potentials, see <cit.>. This program essentially culminates with the fundamental paper of Fritz <cit.>, which at least when the dimension of individual particles is one, essentially provides a complete solution theory for potentials which are twice continuously differentiable everywhere and with compact support.

The solution theory of <cit.> however excludes singular potentials and the construction of solutions is stated therein as an important open problem. In particular it excludes, as both assumptions fail, arguably the most famous potential in 1-dimension, namely the logarithm. Formally, such a system of interacting one-dimensional particles is governed by the following singular stochastic differential equations:

    d𝗑_i(t)=√(2𝔞(𝗑_i(t)))d𝗐_i(t)+𝔟(𝗑_i(t))dt+“2𝔞(𝗑_i(t))∑_j≠ i∂_𝗑_i(t)log|𝗑_i(t)-𝗑_j(t)|dt",

for some nice diffusion 𝔞 and drift 𝔟 functions and with the 𝗐_i being independent standard Brownian motions. The singular force of interaction experienced by the i-th particle 𝗑_i which involves all other particles (𝗑_j)_j≠ i is in quotes because it may need to be renormalised in some way in order to make sense.

An important motivation for studying systems of particles interacting via a logarithmic potential comes from the theory of random matrices <cit.> with the most famous example being the following. If (𝐖_t;t≥ 0) is the Brownian motion on N× N Hermitian matrices then its eigenvalues satisfy the closed system of stochastic differential equations (SDE), called Dyson Brownian motion <cit.>: 

    d𝗑_i(t)=d𝗐_i(t)+∑_j=1,j≠ i^N1/𝗑_i(t)-𝗑_j(t)dt,   i=1,…, N.

This SDE has a unique strong solution with almost surely no collisions between particles for all t>0, even if started from initial conditions with coinciding coordinates, see <cit.>. Dyson Brownian motion, and its variants, has been the object of study for decades. Beyond its intrinsic interest from the perspective of stochastic analysis, probability and integrable systems, it has been a key tool in proving universality for random matrices, see <cit.>.

Coming back to our discussion of infinite systems, as far as we are aware, the only existence (and uniqueness) result of solutions to an ISDE with logarithmic interaction from concrete initial conditions is 
 in the tour-de-force work of Tsai <cit.>. This corresponds to the bulk limit (see the seminal work of Spohn <cit.>) of Dyson Brownian motion (<ref>) and the ISDE is given by (it is also possible to include a parameter β≥ 1 in the drift which corresponds to so-called β-ensembles <cit.>): 

    d𝗑_i(t)=d𝗐_i(t)+β/2lim_k →∞∑_j≠ i:|j-i|≤ k1/𝗑_i(t)-𝗑_j(t)dt,   i∈ℤ.

The initial conditions allowed need to satisfy a certain quantitative “balanced condition": particles' positions are approximately uniform in some averaged sense with lower order corrections, see <cit.>. Moreover, the solution constructed in <cit.> satisfies this property almost surely for all times. The same property is also almost surely satisfied by the β-sine point process <cit.> which is the invariant[We note that the labelled ISDE (<ref>) itself does not have an invariant measure. The simplest possible instance of this phenomenon is that of infinitely many independent Brownian motions (𝗐_i)_i∈ℤ. This process does not have an invariant measure on ℝ^ℤ but when viewed as a process on unlabelled configurations the Poisson point process is invariant for it.] measure of the dynamics if we look at (<ref>) as an unlabelled point process. 

It is also important to mention a remarkable, almost thirty year-long program of Osada, later in collaboration with Tanemura on ISDE, see <cit.>. This program develops a general solution theory and can treat a number of examples, also of other singular interactions and in higher dimensions, but it only yields “near-equilibrium" solutions. These are solutions starting from an abstract, and in particular non-explicit, set of allowed configurations in which the solution stays for all times. This set is of full measure, when viewed as a set of unlabelled configurations, with respect to the invariant[This probability measure on unlabelled configurations is given as data for the problem. One should think of it as playing the role the sine point process plays for the Dyson bulk ISDE (<ref>).] probability measure[For a certain parameter range in our model this measure is infinite and this is another novel feature of the present work.] on point processes. The fact that one deals with an abstract set of initial conditions is not a technical restriction but rather an intrinsic feature of the approach which is based on the theory of Dirichlet forms <cit.>. For other applications of Dirichlet form theory on such particle systems, from a geometric viewpoint, see <cit.>.

Although, as far as we can tell there are no other results on ISDE with logarithmic interaction, there is a great number of results on scaling limits of random matrix dynamics. Possibly the most famous being the edge scaling limit of Dyson Brownian motion to the Airy line ensemble  (𝒜ℒℰ_i(·))_i=1^∞, see <cit.>, a central object in the KPZ universality class <cit.>. To do this, a probabilistic technique, that employs a certain Gibbs resampling property of the paths was developed in <cit.>. This led to an enormous amount of activity and has been extremely successful in showing that various models belong to the KPZ universality class <cit.>. Although our goal in this paper is rather orthogonal to these works there are some interesting connections with a new Gibbs resampling property, see Section <ref>. Finally, the way limits of random matrix dynamics, and related discrete models, were first rigorously studied was through the scaling limit of their space-time correlation functions, see <cit.>. This approach provides delicate quantitative information about the models and beautiful explicit formulae for their correlations, but as far as we can tell cannot be used for the type of results we want to prove. We will survey relevant previous works and how they compare with ours in Section <ref>.

The main purpose of this paper is to analyse an ISDE with logarithmic interaction coming from random matrix dynamics for which we can go beyond what was known previously and to introduce some new ideas for studying the singular drift term and non-intersection of the paths via certain “characteristic polynomials" of the process. We believe that these ideas will be useful in proving analogous results for a class of infinite dimensional dynamics coming from random matrices. The main reason we have chosen to start with this specific model is that it enjoys certain integrable properties which allow us to build on a powerful method of Borodin and Olshanski, see <cit.>, originally developed in the discrete setting of dynamics on partitions. It is plausible that this part of the argument may be replaced by more robust techniques in the future. We have not attempted to do this, partly because the intermediate results we need are of independent interest beyond the setting of ISDE.



 §.§ Main results


The model we consider is the eigenvalue evolution of a diffusion on the space of N× N non-negative definite matrices introduced by Rider and Valko in <cit.>, that we recall in Section <ref>. They used this to prove a matrix analogue <cit.> of Dufresne's identity <cit.>, partly motivated by problems in the theory of stochastic operators related to random matrices, see <cit.>. The N=1 case of this process is a celebrated one-dimensional diffusion that comes up in Matsumoto-Yor's study of exponential functionals of Brownian motion <cit.> and is also connected to random polymers <cit.>. It is much-studied in the applied, statistical and financial mathematics literature <cit.> and sometimes goes by the name inhomogeneous geometric Brownian motion.

The dynamics we consider are given by the following system of stochastic equations, with parameter η∈ℝ,

    d𝔵_i(t) = 𝔵_i(t) d𝗐_i(t) -η/2𝔵_i(t)dt+1/2dt+∑_j=1,j≠ i^N𝔵_i(t)𝔵_j(t)/𝔵_i(t)-𝔵_j(t)dt,   i=1,2,…,N,

with the 𝗐_i being independent standard Brownian motions. By general results <cit.> on such systems in finite dimensions we obtain that (<ref>) has a unique strong non-exploding solution starting from any initial condition in the chamber:

    𝕎_N,+ = {𝐱=(x_1,x_2,…,x_N)∈ℝ^N: x_1≥ x_2 ≥…≥ x_N≥ 0 },

and almost surely, for all positive times lives in the interior of 𝕎_N,+, see Lemma <ref>. It is instructive to see that the SDE (<ref>), by rewriting the interaction term, is given by 

    d𝔵_i(t) = 𝔵_i(t) d𝗐_i(t) -η/2𝔵_i(t)dt+1/2dt+[(-N+1) 𝔵_i(t)+𝔵_i(t)^2∑_j=1,j≠ i^N ∂_𝔵_i(t)log|𝔵_i(t)-𝔵_j(t)|]dt,

which elucidates the fact that the stochastic dynamics (<ref>) correspond to independent one-dimensional diffusions interacting via a logarithmic potential.

In order to state our results precisely we need some notation and definitions. Define the following spaces, endowed with the topology of coordinate-wise convergence,

    𝕎_∞,+   ={𝐱=(x_i)_i=1^∞∈ℝ^ℕ:x_1 ≥ x_2 ≥ x_3 ≥⋯≥ 0 },
    𝕎_∞,+^∘   ={𝐱=(x_i)_i=1^∞∈ℝ^ℕ:x_1 > x_2 > x_3 > ⋯ > 0 }.



In fact, our limiting dynamics will live on an enhanced space containing somewhat more information than the above. It is on this enhanced space, and this is essential, that the dynamics will enjoy a Feller-continuity property. The space is defined as follows.


We define the space

    Ω_+def={ω=(𝐱,γ)∈𝕎_∞,+×ℝ_+: ∑_i=1^∞x_i≤γ},

endowed with the topology of coordinate-wise convergence.   

We observe that, Ω_+ is locally compact, metrizable and separable and the topology can be metrised with the metric 𝖽_Ω_+ as follows, with ω=(𝐱,γ),ω̃=(𝐱̃,γ̃) ∈Ω_+,

    𝖽_Ω_+(ω,ω̃)=∑_i=1^∞|x_i-x̃_i|/2^i(1+|x_i-x̃_i|) + | γ-γ̃|.

Moreover, note that Ω_+ is complete and thus it is a Polish space.





Returning to the dynamics, heuristics related to the hard-edge scaling in random matrix theory <cit.> indicate that we should rescale our stochastic processes in space by 1/N to see non-trivial behaviour. In particular, the rescaled SDE (<ref>) becomes:

    d𝗑_i^(N)(t) = 𝗑_i^(N)(t) d𝗐_i(t) -η/2𝗑_i^(N)(t)dt+1/2Ndt+∑_j=1,j≠ i^N𝗑_i^(N)(t)𝗑^(N)_j(t)/𝗑^(N)_i(t)-𝗑^(N)_j(t)dt,    i=1,2,…,N.


We then embed, for different N ∈ℕ, all these processes on the space Ω_+.


Define the process (𝐗^(N)(t);t≥ 0) on Ω_+ by, with (𝗑^(N)(t);t≥ 0) the solution of (<ref>), 

    𝐗^(N)(t)=((𝗑_i^(N)(t))_i=1^∞,∑_i=1^∞𝗑_i^(N)(t)),   ∀ t ≥ 0,

where 𝗑_i^(N)≡ 0, for i>N.


Finally, we recall the definition of a Feller semigroup and Feller process in Definition <ref>, see <cit.>. For a Polish space 𝔛 write C(ℝ_+,𝔛) for the space of continuous functions on ℝ_+ with values in 𝔛, endowed with the topology of locally uniform convergence. Note that, this is again a Polish space, see <cit.>. We can now state our first main result.

 Let η∈ℝ. Then, there exists a unique Feller semigroup (𝔓_∞(t))_t≥ 0 with associated Feller-Markov process (𝐗_t^Ω_+;t≥ 0)=(((𝗑_i(t))_i=1^∞,γ(t));t≥ 0) on Ω_+ satisfying the following. Let 𝐗_0^Ω_+=ω∈Ω_+ be arbitrary and assume 𝐗^(N)(0)→ω in the topology of Ω_+. Then, as N →∞,

    𝐗^(N)d⟶𝐗^Ω_+ in  C(ℝ_+,Ω_+),

with d⟶ denoting convergence in distribution. In particular, (𝐗_t^Ω_+;t ≥ 0) is a diffusion[A strong Markov process with continuous sample paths.]. Moreover, there exists a coupling of the 𝐗^(N) and 𝐗^Ω_+ on a single probability space such that, almost surely, for any T ≥ 0,

    sup_t ∈ [0,T]∑_i=1^∞(𝗑_i^(N)(t)-𝗑_i(t))^2 N →∞⟶  0.




As far as we can tell, this is the first result on path-space convergence of random matrix dynamics from every single initial condition for which the finite-dimensional dynamics are well-defined. Observe that, the limiting diffusion (𝐗_t^Ω_+;t ≥ 0) can start from configurations with coinciding points. In fact, it can even start from the very singular configurations ((0)_i=1^∞,γ) of having all particles at 0!


It is a natural question whether the limiting process solves an ISDE with logarithmic interaction. Our second main result answers this in the affirmative and in fact gives a lot more information. Throughout this paper we use the notation 𝖫𝖺𝗐(𝐘) for the law of a random variable 𝐘 taking values in a Polish space.

 Let η∈ℝ. Let 𝐱∈𝕎_∞,+^∘ and γ≥∑_i=1^∞ x_i. Consider the Feller process on Ω_+, (𝐗_t^Ω_+;t≥ 0)=(((𝗑_i(t))_i=1^∞,γ(t));t≥ 0) from Theorem <ref> with initial condition 𝐗_0^Ω_+=(𝐱,γ). Then, almost surely, for all t ≥ 0, (𝗑_i(t))_i=1^∞∈𝕎_∞,+^∘ and the (𝗑_i(·))_i=1^∞ is a weak[Recall that a weak solution means that on a (filtered) probability space (𝒱,(𝔙_t)_t≥ 0,𝐕) we can define a (𝔙_t)_t≥ 0-adapted sequence of independent standard Brownian motions (𝗐_i)_i=1^∞ and the process (𝗑_i)_i=1^∞  which is adapted with respect to (𝔙_t)_t≥ 0 such 𝐕-a.s. (<ref>) holds for all t≥ 0; in particular the pair ((𝗑_i)_i=1^∞,(𝗐_i)_i=1^∞) is considered a weak solution to (<ref>), see <cit.>. ]
solution to the following ISDE with logarithmic interaction starting from 𝐱, namely it satisfies,


    𝗑_i(t)=x_i+∫_0^t 𝗑_i(s)d𝗐_i(s) -η/2∫_0^t𝗑_i(s)ds+∫_0^t∑_j=1,j≠ i^∞𝗑_i(s)𝗑_j(s)/𝗑_i(s)-𝗑_j(s)ds,   i ∈ℕ,

where the (𝗐_i(t);t≥ 0) are independent standard Brownian motions. Moreover, if we denote by (𝐗(t;γ);t≥ 0) the solution above corresponding to γ, then

    𝖫𝖺𝗐(𝐗(·;γ))≠𝖫𝖺𝗐(𝐗(·;γ̃)), whenever γ≠γ̃.

Finally, out of these solutions there exists a unique one such that almost surely t ↦∑_i=1^∞𝗑_i(t;γ) is continuous for all t ≥ 0 given by the choice γ=∑_i=1^∞ x_i
and this solution is a Markov process.


Observe that, the singular interaction drift in (<ref>) is really a renormalised[Observe that, without subtracting the diverging counterterm the logarithmic interaction is clearly infinite.] logarithmic interaction term, since

    ∑_j=1, j≠ i^∞𝗑_i(t)𝗑_j(t)/𝗑_i(t)-𝗑_j(t)=lim_N→∞[(-N+1)𝗑_i(t)+𝗑_i(t)^2∑_j=1, j≠ i^N 1/𝗑_i(t)-𝗑_j(t)].

In some sense, we could interpret the dynamics (<ref>) as geometric/exponential Brownian motions interacting via a renormalised logarithmic potential. As far as we know, this result is the first construction of solutions to an ISDE with logarithmic interaction from every single initial condition for which the infinitesimal singular drift term makes sense at time t=0. It is optimal since we need the condition ∑_i=1^∞ x_i<∞ for otherwise the drift term is infinite. Also, we note that, the initial configurations, and more generally the distribution of the dynamics at time t≥ 0, do not need to resemble in any qualitative sense the invariant measure (when it exists) to be defined below, like they do, to some extent, in the case of the bulk Dyson ISDE <cit.>. This result is also the first, as far as we can tell[As far as we can tell, even for the class of initial conditions considered in the works <cit.>, a single solution is first constructed which is then shown (under additional assumptions) to be unique.], concrete construction of non-unique (even from a single initial condition) solutions for ISDE with logarithmic interaction. This is an explicit illustration of a phenomenon that is not seen in finite-dimensional random matrix dynamics. 

We now move to our results on the invariant measure and convergence to equilibrium for the dynamics when η>-1. Note that, this restriction is also required for the finite-dimensional dynamics (<ref>) for otherwise there is no invariant probability measure. In fact, the construction of solutions to the ISDE with logarithmic interaction above, when η≤ -1, appears to be the first, of any kind, for which there is no invariant probability measure when viewed as an evolution on unlabelled point processes. We need some more notation and terminology. Let ℐ⊂ℝ be a union of open intervals. Let 𝖢𝗈𝗇𝖿(ℐ) denote the space of configurations over ℐ, namely locally finite collections of points in ℐ, or equivalently the space of ℤ_+-valued measures on ℐ endowed with the vague topology, see <cit.> for details. A determinantal point process on ℐ, with correlation kernel 𝔎, is a probability measure on 𝖢𝗈𝗇𝖿(ℐ) which is determined[Subject to certain mild conditions so that the correlation functions determine the point process, see <cit.>.] by the fact that all its correlation functions (ρ_n)_n=1^∞ with respect to Lebesgue measure are given as ρ_n(x_1,x_2,…,x_n)=(𝔎(x_i,x_j))_i,j=1^n, see <cit.> for rigorous details. 



Let η>-1. The inverse Bessel point process 𝖨𝖡𝖾𝗌^Conf_η with parameter η is the determinantal point process on (0,∞) with correlation kernel 𝖪_η given by 

    𝖪_η(x,y)=
    8/xy𝕁_η(8/x,8/y),   x,y ∈ (0,∞),

where 𝕁_η(x,y) is the so-called the Bessel kernel, with J_η the Bessel function of order η,

    𝕁_η(x,y)=
    x^1/2J_η+1(x^1/2)J_η(y^1/2)-
    y^1/2J_η+1(y^1/2)J_η(x^1/2)
    /2(x-y).



𝖨𝖡𝖾𝗌^Conf_η is called the inverse Bessel point process because it simply consists of the inverse points (up to multiplicative constant) of the Bessel determinantal point process with correlation kernel 𝕁_η <cit.>. The Bessel point process is the universal limit arising at the hard edge of random matrices, see <cit.>.


Since 𝖨𝖡𝖾𝗌^Conf_η consists of distinct points it gives rise to a unique probability measure  𝖨𝖡𝖾𝗌_η on 𝕎_∞,+^∘ by labelling the points of 𝖨𝖡𝖾𝗌^Conf_η in a decreasing order. Define the space

    Ω_+^0def={ω=(𝐱,γ)∈Ω_+:∑_i=1^∞x_i=γ},

and write π:Ω_+ →𝕎_∞,+ for the map given by π((𝐱,γ))=𝐱. 


Let η>-1. Define 𝔐^η to be the unique Borel probability measure on Ω_+ which is supported on Ω_+^0 and satisfies 

    π_*𝔐^η=𝖨𝖡𝖾𝗌_η.



Then, we have the following result on convergence to equilibrium for the process (𝐗_t^Ω_+;t ≥ 0) on Ω_+.

 
Let η>-1 and 𝔎 a Borel probability measure on Ω_+. Consider the Feller process (𝐗_t^Ω_+;t ≥ 0) constructed in Theorem <ref> and suppose 𝖫𝖺𝗐(𝐗_0^Ω_+)=𝔎. Then, as t→∞,

    𝐗_t^Ω_+d⟶𝐙,   where 𝖫𝖺𝗐(𝐙)=𝔐^η.

In particular, if η>-1, 𝔐^η is the unique invariant measure of (𝐗_t^Ω_+;t ≥ 0).


As far as we can tell, the result above is the only convergence to equilibrium result on ISDE with logarithmic interaction. Nevertheless, there is very interesting recent work on ergodicity of unlabelled infinite-dimensional diffusions (namely looking at diffusions on the space 𝖢𝗈𝗇𝖿) with logarithmic interaction. Dyson's model in the bulk is considered in <cit.> and a more general class of models (including the β-ensemble version (<ref>) of Dyson's model) is treated by Suzuki in <cit.>. Both <cit.> and <cit.>  use  a Dirichlet form approach. The paper <cit.> builds on Osada-Tanemura's theory and <cit.> makes use of the DLR equations <cit.> for the β-sine point process <cit.>.



We finally prove the following result which says that although we can construct infinitely many non-equal in law solutions for the ISDE (<ref>) starting from the inverse Bessel points 𝖨𝖡𝖾𝗌_η, there is in some sense only one true equilibrium solution. Another interesting feature is that all of these different solutions start distributed according to 𝖨𝖡𝖾𝗌_η, all but one of them will stop being distributed according to 𝖨𝖡𝖾𝗌_η instantaneously (this does not appear in the statement below but will be clear from the proof), but all of them will converge in distribution back to 𝖨𝖡𝖾𝗌_η as t→∞.


Let η>-1. Let 𝔪 be a Borel probability measure on Ω_+ satisfying π_*𝔪=𝖨𝖡𝖾𝗌_η. Consider the Feller process (𝐗_t^Ω_+;t≥ 0) with random initial condition 𝖫𝖺𝗐(𝐗_0^Ω_+)=𝔪 and write (𝐗_t^Ω_+;t≥ 0)=((𝐗(t;𝔪),γ(t;𝔪));t≥ 0) where 𝐗(t;𝔪)=(𝗑_i(t;𝔪))_i=1^∞. Then, (𝗑_i(·;𝔪))_i=1^∞ solves (<ref>) with initial condition distributed according to 𝖨𝖡𝖾𝗌_η. Out of these solutions (for different 𝔪) there exists a unique solution 𝐗(·;𝔪) satisfying

    𝖫𝖺𝗐(𝐗(t;𝔪))=𝖨𝖡𝖾𝗌_η,   ∀ t ≥ 0,

and it is given by the choice 𝔪=𝔐^η.









 §.§ Strategy of proof

(a) Convergence on path-space and Feller property. Our starting point is a certain non-obvious consistency relation the finite-dimensional dynamics enjoy when N varies. Let (𝔓_N(t))_t≥ 0 be the Markov semigroup associated to the unique strong solution of (<ref>). For 𝐱=(x_1>x_2>⋯>x_N+1), we consider the Markov kernel, 

    Λ_N^N+1(𝐱,d𝐲)=N!∏_1≤ i<j ≤ N(y_i-y_j)/∏_1≤ i<j ≤ N+1(x_i-x_j)1_x_1>y_1>x_2>⋯>x_N>y_N>x_N+1d𝐲.

This Markov kernel has an important interpretation in terms of Hermitian random matrices whose law is invariant under unitary conjugation <cit.>  and can be extended to 𝐱 with coinciding coordinates. We will say more in Section <ref>. The consistency relation we are alluding to, is the following so-called intertwining, between the semigroups:

    𝔓_N+1(t)Λ_N^N+1=Λ_N^N+1𝔓_N(t),   ∀ t≥ 0,  N∈ℕ.

We give a direct proof of this relation but also a different argument using matrix stochastic calculus which gives more intuition regarding where it actually comes from. It is worth mentioning that Dyson Brownian motion also satisfies a relation of this kind, see <cit.>.
 

Now, one can view (𝕎_N,+,Λ_N^N+1)_N=1^∞ as a projective system of measures or equivalently as a Markov chain moving in discrete time, see <cit.> for background. Then, it is a classical theorem of Pickrell <cit.> and Olshanski-Vershik <cit.>, in equivalent form, that the entrance boundary of this Markov chain can be identified with the space Ω_+ and the extremal entrance laws Λ_N^∞(ω,d𝐱), with ω∈Ω_+, have a random matrix interpretation. Moreover, it can be shown that all Markov kernels appearing above are Feller continuous. Then, by the method of intertwiners of Borodin and Olshanski <cit.> the above considerations guarantee the existence of a Feller semigroup (𝔓_∞(t))_t≥ 0 and associated Feller process 𝐗^Ω_+ on Ω_+ which is uniquely determined via the intertwinings:

    𝔓_∞(t)Λ_N^∞=Λ_N^∞𝔓_N(t),  ∀ t ≥ 0,  N ∈ℕ.


To prove convergence to equilibrium in the long-time t limit for 𝐗^Ω_+, Theorem <ref>, we extend the method of intertwiners to deal with such questions by proving the following simple statement. If the finite-dimensional processes converge to equilibrium from any initial condition then the infinite-dimensional process converges to equilibrium from any initial condition. This holds in a general setting and can be applied to other models.


Coming back to describing the Feller process 𝐗^Ω_+, at this stage we know essentially nothing about its trajectories. We do not know if and how it is approximated by the finite dimensional dynamics. We do not know if it has continuous sample paths. It might even be, in some way, degenerate[This may well happen. The most famous example satisfying (<ref>), Dyson Brownian motion itself, also gives rise, by the method of intertwiners, to a Feller process on an extension Ω of the space Ω_+, see Section <ref>, but this “infinite-dimensional stochastic process" is a degenerate deterministic flow on the coordinates of Ω.].

Proving convergence on the path-space C(ℝ_+,Ω_+) would resolve these questions. Since we are dealing with Feller processes one can prove this convergence by proving convergence at the level of the infinitesimal generators for a sufficiently large class of functions. This is in general highly non-trivial though since we are dealing with infinite-dimensional objects. But we have more structure which helps. Let us write 𝖫_N and 𝖫_∞ for the infinitesimal generators of the Feller semigroups (𝔓_N(t))_t≥ 0 and (𝔓_∞(t))_t≥0 respectively. First of all, if 𝒞_N is a core for 𝖫_N for each N ∈ℕ, the space of functions

    span(⋃_K=1^∞Λ_K^∞𝒞_K),

is known to be a core for 𝖫_∞. In our setting, 𝒞_K will be a class of smooth functions with compact support with symmetry, see Section <ref>. Then, for an arbitrary 𝐟 in (<ref>), of the form 𝐟=Λ_K^∞ g, for g ∈𝒞_K, we will show that there exists 𝐟_N in the domain of 𝖫_N such that we have the following convergence in a uniform sense (we are abusing notation here; to be precise we have to embed all of them on Ω_+), as N→∞,

    𝐟_N    ⟶𝐟, 
    𝖫_N 𝐟_N    ⟶𝖫_∞𝐟.

The right choice for 𝐟_N will be nothing but,

    𝐟_N=Λ_N-1^N Λ_N-2^N-1⋯Λ_K^K+1g.

Thus, showing (<ref>) will turn out to be equivalent to proving 

    sup_𝐱^(N)∈𝕎_N,+|Λ_K^∞g(ω(𝐱^(N)))-Λ_N-1^NΛ_N-2^N-1⋯Λ_K^K+1g(𝐱^(N))|N→∞⟶ 0,

where ω(𝐱^(N)) denotes a certain embedding of 𝐱^(N)∈𝕎_N,+ into Ω_+, see Section <ref>. 

This is what we call the uniform approximation theorem and it is arguably our main contribution for this part of the argument. It is inspired by an analogous theorem in the combinatorial setting of the Gelfand-Tsetlin graph <cit.>. There are some important new difficulties which arise in the continuous setting of random matrices however. First of all, the explicit formulae for Λ_K^∞ and Λ_N-1^N⋯Λ_K^K+1 as K× K determinants, due to Olshanski-Vershik <cit.> and Olshanski <cit.>, that we start with, when extended to arbitrary ω∈Ω_+ and 𝐱∈𝕎_N,+ involve as entries derivatives of distributions which is problematic. Fortunately, we can make sense of them by virtue of the fact that we are testing[There is a tension here between the need for the function spaces 𝒞_K to be large enough to deduce convergence from them while at the same time they should only include nice enough functions for the explicit formulae to make sense. We note that, there are alternative, random matrix interpretations of Λ_K^∞ and Λ_N-1^N⋯Λ_K^K+1, see Section <ref>, which can be tested against arbitrary bounded Borel functions but, as far as we can tell, these cannot be used to prove (<ref>), or even guess that a statement like (<ref>) is true.] against compactly supported symmetric (which is important) smooth functions g∈𝒞_K. Moreover, in the combinatorial setting once one gets a good explicit formula as a K× K determinant for the analogues of Λ_K^∞ and Λ_N-1^N⋯Λ_K^K+1 the asymptotic analysis is fairly straightforward: one needs to take the limit of each individual entry in the determinant. In the random matrix setting, despite some post-processing of the formulae of <cit.> to make them look as similar as possible, this does not appear to be the case. For this reason we needed to devise a non-trivial inductive argument in order to establish the uniform asymptotics (<ref>). Finally, to prove (<ref>), by virtue of the intertwining (<ref>), it again boils down to (<ref>).
Hence, given (<ref>) and (<ref>), by general results <cit.>, we obtain the path-space convergence in (<ref>). Some additional arguments then also give convergence in ℓ^2 as in (<ref>). 


It is interesting to note that in the above argument we did not need to use any explicit information about the generator 𝖫_∞. This should not be too surprising. The whole point, and in some sense appeal, of the method of intertwiners is that at the level of dynamics we only need to work with 𝖫_N for fixed N. However, this is also a caveat. If one wants to understand the actual dynamics of 𝐗^Ω_+ new arguments are needed. 

(b) Non-intersection of the paths. The next key step is to show non-intersection of the limiting paths (𝗑_i(·))_i=1^∞. This is essential input to obtain the ISDE. It is also essential input in the approach of Tsai <cit.> and Osada-Tanemura <cit.>; in Tsai's work proving this is one of the major challenges, see discussion around Remark 2.9 therein. As far as we can tell, the method of intertwiners is not useful for proving this type of statement. There is also a new Gibbs resampling property behind the SDE (<ref>), that we attempted to use, but this approach runs into problems that we explain in Section <ref>. So we turned back to make use of the stochastic equations. This cannot be done directly in limit: we have no stochastic equations for the infinite-dimensional process to speak of yet! Instead, we need to control, uniformly in N, the probabilities that the finite-dimensional paths solving (<ref>) come close directly from the SDE. As far as we can tell, this is the first[Both Tsai's <cit.> and Osada-Tanemura's <cit.> approaches for non-intersection do not make direct use of the finite-dimensional SDE. Similarly, the Gibbsian line ensemble approach <cit.> to prove non-intersection is totally different and does not use SDE in any way.] time this has been done. Of course, for fixed N, showing non-intersection is standard and this is the mainstream approach to attack Dyson-like SDE, see <cit.>, but the bounds in the literature blow up with N. There is good reason for this, that we comment on below.


In essence, although the actual proof is presented very differently in terms of multiple stopping times, we establish statements which informally can be interpreted as the following. Let 𝐱^(N) be arbitrary in the interior of 𝕎_N,+ such that ω(𝐱^(N))⟶ω=(𝐱,γ), with 𝐱∈𝕎_∞,+^∘. Then, for any n ∈ℕ, T≥ 0 and ε>0, we can find δ such that,

    sup_N≥ n+1Prob(inf_t∈ [0,T]|𝗑_n^(N)(t)-𝗑_n+1^(N)(t)|<δ|𝗑^(N)(0)=𝐱^(N))< ε.


The reason the bounds, and observables used to obtain them, that are given in the literature for fixed N, are not useful as N →∞, is that they are controlling the collision times of all N paths simultaneously and this cannot work. Instead we perform a double induction argument to prove the statements above one pair of paths at a time. The main ingredients are certain novel observables or Lyapunov functions associated to the dynamics given by variations of the reverse characteristic polynomial of a matrix with eigenvalues (𝗑^(N)_i(t))_i=1^N, see the functions f_n^N and Ψ_n^N and discussion around (<ref>). The crux of the argument, and the reason considering the aforementioned observables is actually useful is the following. When applying Itô's formula to them, the term which comes from the singular drift, which would be hard to control uniformly in N, has negative sign and can simply be dropped to give an N-independent bound. This feature is rather generic. For example the argument works for the β-ensemble version[Include a multiplicative constant β in front of the singular interaction drift.] of the dynamics and variations of it should work for other models. Even if we hadn't known convergence of the finite-dimensional dynamics, the argument gives that any possible subsequential limit would necessarily consist of non-intersecting paths.

(c) The ISDE. Finally, to prove (𝗑_i(·))_i=1^∞ solves the ISDE (<ref>), modulo technicalities, this boils down to convergence of the singular drift term in the finite-dimensional SDE to its infinite-dimensional counterpart. Again, viewing things in terms of certain “characteristic polynomials" of the process will do the trick. In particular, if we write

    Φ_i^N(z;𝐱^(N))=∏_j=1,j≠ i^N(1-x_j^(N)z)^2,  𝐱^(N)∈𝕎_N,+,

then we can observe that,

    ∑_j≠ i^N 𝗑_i^(N)(t)𝗑_j^(N)(t)/𝗑_i^(N)(t)-𝗑_j^(N)(t)=1/2d/d zlogΦ_i^N(z,𝗑^(N)(t))|_z=(𝗑_i^(N)(t))^-1.

Then, making use of the following fact, by virtue of convergence in Ω_+, where convergence holds uniformly on compact sets in z∈ℂ,

    Φ_i^N(z;𝗑^(N)(t)) ⟶  e^-2(γ(t)-𝗑_i(t)) z∏_j=1,j≠ i^∞e^2𝗑_j(t)z(1-𝗑_j(t)z)^2,

combined with some further arguments that make use of the fact that paths are non-intersecting, and no 𝗑_i hits zero, we can show convergence of the drift term.

At this stage the ISDE for the (𝗑_i(·))_i=1^∞ coordinates may still depend on γ(·). Then, the final part of the argument, making use of the structure of the ISDE shows that almost surely, for all t>0, ∑_i=1^∞𝗑_i(t)=γ(t) which completes the proof. Thus, in some sense, the true state space of the diffusion 𝐗^Ω_+ is Ω_+^0 and Ω_+ \Ω_+^0 acts as a kind of entrance boundary for it. At this point all the hard work is done and the proofs of the remaining statements in our main results are fairly straightforward.






 §.§ Previous approaches


The kinds of problems we are considering have been studied intensely for decades. As a result a number of rather sophisticated approaches have been developed to attack them. We survey them below. To different extents, we have attempted to apply each of them to our problem.

(1) Tsai's important paper <cit.> was the only, up until now, work which gave existence of solutions to an ISDE with logarithmic interaction from explicit initial conditions. Tsai uses in a very essential and delicate way the translation-invariance of the dynamics and a certain novel monotonicity property of the corresponding process of gaps between particles. The translation-invariance is definitely not true for our model and it is unclear what the right analogue of the monotonicity property (if one exists) should be. For these fundamental reasons it seems hard to adapt the approach of <cit.> to even give solutions  from a class of initial conditions in our case.

(2) We now move to Osada and Tanemura's program <cit.>. This spans almost three decades and several papers so it is impossible to survey in a few lines. The upshot is the following: one is given as basic data a probability measure 𝖬 on the configuration space 𝖢𝗈𝗇𝖿(𝒵) of some Euclidean space 𝒵 consisting of unlabelled collections of points. From this, under certain assumptions, one first constructs a diffusion on 𝖢𝗈𝗇𝖿(𝒵), starting from any initial condition in an abstract set 𝒮_𝖬⊂𝖢𝗈𝗇𝖿(𝒵),  so that 𝖬(𝒮_𝖬)=1, with 𝖬 as invariant measure. With this as input, and under further assumptions, a weak solution to a labelled ISDE is constructed (always starting and remaining in a labelled version of 𝒮_𝖬), which under additional conditions is shown to be the unique one satisfying certain properties. For some of these conditions[These are for the most part conditions on the correlation functions of 𝖬 and its Campbell measures <cit.>.] black-box theorems already exist to verify them, however some of them do need to be checked by hand. This is in general a highly non-trivial task. For example, the main purpose of the long preprint <cit.> is to verify these conditions in the case of the ISDE associated to the stationary version of the Airy line ensemble in order to apply the theory.

Let us be more precise on how we think the Osada-Tanemura theory relates to our approach. We believe that if one attempts (for η>-1 certain modifications are needed but we will not check the technical conditions here, while for η≤ -1 a fundamental extension of this theory is required since there is no invariant probability measure on 𝖢𝗈𝗇𝖿((0,∞)) to start with) to use this theory to construct “near-equilibrium" solutions for our model, these should match, in a way that needs to be made precise, for initial conditions 𝐱∈𝕎_∞,+^∘ that qualitatively resemble the invariant measure, the solutions that arise from Ω_+^0. It seems plausible that this is a more general phenomenon. The Dirichlet form approach will pick out a distinguished solution from the ones living on an enhanced space. Another related point is that the singular interaction drift of the ISDE in Osada-Tanemura's theory is given by the so-called logarithmic derivative of the point process <cit.>. In some sense this notion associates to the invariant point process a canonical limiting characteristic polynomial (a random analytic function). It would be interesting to understand these potential connections more precisely and rigorously in the future.

(3) The recent work of Landon <cit.> on the edge scaling of β-Dyson Brownian motion at equilibrium is also worth mentioning. Using a different approach to previous works, that takes rigidity estimates for the invariant point process (the β-Airy point process <cit.>) as a key ingredient, Landon can prove convergence under a subsequence for the equilibrium dynamics for all β≥ 1 (the paper <cit.> only deals with β=1,2,4); the fact that the process is in equilibrium is essential for the use of rigidity estimates. However, the limiting process is not shown to solve an ISDE, or even have non-intersecting paths. Nevertheless, it is very natural to expect that these statements should be true.

(4) A different method for proving path-space convergence of random matrix dynamics, having special structure, is the probabilistic approach of Gibbsian line ensembles <cit.>. This type[We will not give the precise definitions here, see for example <cit.>.] of Gibbs property is, in some sense, the right dimension-independent analogue of the fact that the N-dimensional Dyson Brownian motion (<ref>) can equivalently be constructed as N independent Brownian motions conditioned to never intersect <cit.>. This approach is  useful for proving tightness of the paths. To conclude convergence some more information is needed. The current state of the art, in the case of models in the KPZ universality class, is the recent remarkable strong characterisation theorem <cit.> of the Airy line ensemble. This says that, beyond a Brownian resampling property, we only need to know that the top path approximates a parabola to pin down the Airy line ensemble.

(5) Yet another approach to taking limits of random matrix dynamics is to view the N-dimensional interacting SDE as diffusions on 𝖢𝗈𝗇𝖿 and take the limit of their space-time correlation functions. The state of the art, for limits of Dyson Brownian motion in the bulk and non-intersecting squared Bessel processes starting from a class of initial conditions, appears to be the work of Katori-Tanemura <cit.>. As far as we can tell, this approach cannot, on its own at least, be used to obtain results on ISDE.

(6) We now comment on the method of intertwiners[For uses of intertwining relations, with other applications in mind, see for example <cit.>.]. An exposition can be found in the lecture notes and ICM proceedings by Olshanski <cit.>. The method was originally introduced in a discrete setting to construct infinite-dimensional Feller processes coming from continuous-time jump dynamics on partitions which satisfy an intertwining. These Markov dynamics can be thought of as discrete variants of finite-dimensional SDE with logarithmic interaction. The main example the method was developed for in <cit.> is related to the projective system associated to the Gelfand-Tseltin graph <cit.>, whose boundary can be identified with an infinite-dimensional space 𝔊𝔗 with continuous parameters, akin to Ω_+. The invariant measure of the process, the analogue of 𝔐^η, is given by a so-called zw-measure which comes up in representation theory and harmonic analysis <cit.>. One could think of these measures as a more sophisticated version of determinantal point processes that arise in random matrix theory. The current state of the art on this construction is the paper <cit.> of Olshanski, where the generator of the process on 𝔊𝔗 is computed explicitly, not in the natural coordinates of 𝔊𝔗, but rather in a different set of variables and is realised by a second order differential operator. The main idea is that in this other set of variables, which come from abstract algebraic considerations, the generator is not singular. Nevertheless, it is still not known if the process of <cit.> has continuous[Of course, by the Feller property it has a modification with càdlàg paths <cit.>.] sample paths  and this is one of the questions posed in <cit.>. In fact, it is not unreasonable, in analogy with the results of our paper, to expect even more: that in natural coordinates this process should solve some kind of ISDE with singular interaction, at least for some initial conditions. This is probably a hard problem.


In the random matrix setting the method of intertwiners was first applied by one of us in <cit.>. There an abstract construction of an infinite-dimensional Feller process related to the Cauchy ensemble <cit.> is provided. We recall the main result of <cit.> in Theorem <ref> below and we extend it, using the machinery we develop in this paper, to a strong path-space convergence result which also proves that the limiting process has continuous sample paths.


(7) Finally, let us mention that for a wide class of finite-dimensional systems with interactions of the form  (<ref>) a complete solution theory exists, see <cit.>. In finite dimensions one can even construct dynamics when collisions occur between particles (this corresponds to β<1 in the Dyson SDE) using the theory of multivalued SDE, see <cit.>. As far as we can tell, nothing is known for the corresponding question in infinite dimensions.




 §.§ Open problems

We end this introduction with a number of open problems.

    
  * Gibbs property. It would be very interesting if the limiting non-intersecting paths (𝗑_i)_i=1^∞ satisfy a Gibbs resampling property <cit.>. We say more about this problem in Section <ref>.
    
  * Space-time determinantal correlations. A special class of finite-dimensional dynamics with logarithmic interaction, including (<ref>), starting from any deterministic initial condition, is known to have determinantal space-time correlations, see <cit.>. It is an interesting question whether the limiting process we constructed enjoys the space-time determinantal property with an explicit correlation kernel. A somewhat easier question, which nevertheless involves the same fundamental difficulties, is to try to do this for the dynamics starting from the equilibrium measure.
    
  * Rate of convergence to equilibrium. It would be interesting if one could quantify (in any sense) the convergence to equilibrium from Theorem <ref>.
    
  * Existence of solutions for other models. 
    Our approach to construct solutions to ISDE with logarithmic interaction will also work for the dynamical Cauchy model in  Section <ref>. There is an additional and non-trivial complication there in that the pre-limit paths can cross zero. Towards a more general class of models, one needs to replace the first, integrable, part of our argument with more robust techniques coming from stochastic analysis or probability. Then[In fact, most likely an extension of the second part of our argument could be used to replace certain aspects of the first. At least when it comes to tightness of the paths, a uniform bound like (<ref>) is the main ingredient in <cit.>. Note that, if one only cares about the ISDE we do not need the full power of a result like Theorem <ref>.], we believe variations of the second part of our argument can be applied.
    
    
  * Uniqueness of solutions. We proved a certain kind of uniqueness in Theorem <ref>. However, the more natural question is to prove uniqueness, for a general class of ISDE with singular interaction, within a space of paths that take values in certain “rigid configurations", the analogue of “balanced configurations" from the work of Tsai for the Dyson ISDE <cit.>. The precise meaning of “rigid configurations" will need to be model-specific. One would hope that in such spaces of paths the tail-end of the singular drift term can be controlled which is the key requirement for uniqueness. Finally, one needs to ensure solutions taking values in such spaces exist in the first place.




  
Organisation of the paper In Section <ref>, we first recall the method of intertwiners of <cit.> in a general setting. We then extend it to deal with convergence to equilibrium and most importantly we prove the uniform approximation theorem and its various consequences on consistent Feller processes. In Section <ref>, we establish various properties of the solution to the finite-dimensional SDE (<ref>) that allow us to apply the results of Section <ref>. We also briefly discuss a new Gibbs resampling property for (<ref>). In Section <ref>, we first show non-intersection of the limiting paths in Theorem <ref> and then obtain the ISDE in Theorem <ref>. In Section <ref>, we quickly apply our framework to a dynamical model for the Cauchy ensemble, upgrading the abstract results of <cit.>.



  
Acknowledgements TA is grateful to Duncan Dauvergne and Evgeni Dimitrov for a useful discussion regarding the Gibbs resampling property and to Kohei Suzuki for very informative discussions on the Dirichlet form approach to infinite dimensional diffusions with singular interaction. Part of this research was performed while TA was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation (Grant No. DMS-1925919). ZSM is grateful to EPSRC for funding through a PhD studentship.
   
 

§ CONVERGENCE OF CONSISTENT MARKOV PROCESSES

The main purpose of this section is to establish a general result for path space convergence for a sequence of Markov processes related to random matrices  satisfying  consistency relations. 




 §.§ Method of intertwiners: Construction of the Feller process


We recall the method of intertwiners of <cit.>. Since there is a subtle difference in that we put the weak topology on spaces of probability measures instead of the topology induced by the total variation norm used in <cit.> we give a detailed exposition and some proofs. To set things up, we will need to collect several definitions and facts which can be found, for example, in <cit.>.

The measurable structures on topological spaces below are always understood to be the Borel σ-algebras. A Polish space 𝒳 is a separable completely metrisable topological space. A standard Borel space is a measurable space (𝒳,ℱ) such that there exists a metric on the base set so that the induced Borel σ-algebra is ℱ. A Borel isomorphism is a bi-measurable bijective function between two standard Borel spaces. 

We write ℳ_p(𝒳) for the set of Borel probability measures on a Polish space 𝒳. Unless otherwise stated, we always endow ℳ_p(𝒳) with the weak topology. This makes it a Polish space (and so a standard Borel space). Moreover, the Borel σ-algebra coincides with the σ-algebra generated by the evaluation maps μ↦μ(𝒜), for 𝒜 measurable (but the topology on the base set ℳ_p(𝒳) generated by these maps is different from the weak topology). 

For a Polish space 𝒳 denote by ℬ(𝒳) the space of real-valued bounded Borel functions on 𝒳. A Markov kernel 𝒦: 𝒳_1 →𝒳_2 from 𝒳_1 to 𝒳_2 is a function 𝒦(x,𝒜), where x∈𝒳_1 and 𝒜⊂𝒳_2 is a measurable subset, such that 𝒦(x,·) is a probability measure on 𝒳_2 and x↦𝒦(x,𝒜) is a measurable function on 𝒳_1. Observe that, a Markov kernel 𝒦: ℰ_1 →𝒳_2 can be viewed as a map on probability measures 𝒦:ℳ_p(𝒳_1)→ℳ_p(𝒳_2) or as a map on functions 𝒦:ℬ(𝒳_2) →ℬ(𝒳_1) in the obvious way.

For a locally compact Polish space 𝒳, we write C_0(𝒳) for the space of continuous (real-valued) functions vanishing at infinity, endowed with the supremum norm which makes it a Banach space.
 
 Let 𝒳_1 and 𝒳_2 be locally compact Polish spaces. A Markov kernel 𝒦:𝒳_1 →𝒳_2 is Feller if the induced map from ℬ(𝒳_2) to ℬ(𝒳_1) maps C_0(𝒳_2) to C_0(𝒳_1).


The following is our basic data. Suppose that we are given:

    
  * A sequence of locally compact Polish spaces (ℰ_N)_N=1^∞.
    
  * For each N∈ℕ, a Feller Markov kernel ℒ_N^N+1:ℰ_N+1→ℰ_N.



We would now like to associate a certain limit object to the above data. There are a number of (not always equivalent) ways of doing this <cit.>; see in particular Section 4.3.1 of <cit.> for the connection to entrance boundaries of Markov chains mentioned in the introduction. Below, we discuss the bare minimum we need for our applications.

To begin with, observe that for any N ∈ℕ, the affine maps ℒ_N^N+1:ℳ_p(ℰ_N+1) →ℳ_p(ℰ_N) are continuous. Hence, we can take the projective limit of the convex topological spaces ℳ_p(ℰ_N), which is defined as follows, see Section 3 of <cit.>,

    lim_←Topℳ_p(ℰ_N)def={(μ_N)_N=1^∞∈∏_N=1^∞ℳ_p(ℰ_N): μ_N+1ℒ_N^N+1=μ_N,  ∀ N ∈ℕ}.

This is endowed with the projective limit topology. It is in fact a Polish simplex (and in particular a standard Borel space), see Section 4 of <cit.>. The following terminology will be used throughout the paper.

We call any element of (<ref>) a consistent sequence of (probability) measures.



We then come to the following definition.

Suppose that ℰ_∞ is a locally compact Polish space, equipped with Feller Markov kernels ℒ_N^∞:ℰ_∞→ℰ_N satisfying, for all N ∈ℕ, ℒ_N+1^∞ℒ_N^N+1=ℒ_N^∞, so that the map,

    ℳ_p(ℰ_∞)    →lim_←Topℳ_p(ℰ_N)
    𝔪   ↦(𝔪ℒ_N^∞)_N=1^∞,

is a Borel isomorphism. Then, we say that ℰ_∞ is a Feller boundary of the system (ℰ_N,ℒ_N^N+1)_N=1^∞.


We will not be concerned with existence and uniqueness of such an object for general systems here. In related settings, abstract existence and uniqueness (in an appropriate measurable sense) results exist, see <cit.>. For our ultimate purposes of constructing dynamics on ℰ_∞, the above abstract setting is only useful if we have a nice explicit description for it and the kernels ℒ_N^∞. This is in general a difficult problem but in some cases it admits a good solution. This is the case for the concrete applications to random matrix dynamics we have in mind.

We can now prove the following lemma, which is essentially Lemma 2.3 of <cit.>.

Assume ℰ_∞ is a Feller boundary of (ℰ_N,ℒ_N^N+1)_N=1^∞. Then, the space 

    span(⋃_K=1^∞ℒ_K^∞C_0(ℰ_K))

is dense in C_0(ℰ_∞).



On a locally compact Polish space 𝒳, let ℳ(𝒳) denote the set of signed measures with finite total variation norm ||· ||_tv.
By the Riesz representation theorem the Banach space (ℳ(ℰ_∞),||· ||_tv) is the Banach dual of C_0(ℰ_∞). Hence, in order to prove the result it suffices to show that if μ∈ℳ(ℰ_∞) annihilates all functions in (<ref>) then necessarily μ = 0.
Observe that, this last statement makes no reference to the topology we put on ℳ(ℰ_∞). Assume that μ∈ℳ(ℰ_∞)  annihilates ℒ_N^∞ C_0(ℰ_N), for all N ∈ℕ. This is equivalent, again by the Riesz representation theorem applied to each C_0(ℰ_N), to μℒ_N^∞∈ℳ(ℰ_N) being 0, for all N ∈ℕ. We can write μ(·)=||μ_+||_tvμ_+(·)-||μ_-||_tvμ_-(·) for μ_+,μ_- ∈ℳ_p(ℰ_∞). Since μℒ_N^∞ =0, for all N ∈ℕ we obtain ||μ_+||_tv=||μ_-||_tv and thus μ_+ℒ_N^∞=μ_-ℒ_N^∞, for all N ∈ℕ. Since the map (<ref>) is a bijection we must have μ_+=μ_- and hence μ =0 as desired.


Before constructing Feller Markov dynamics on ℰ_∞, we need to recall the definition of a Feller semigroup, see <cit.>.


 Let 𝒳 be a locally compact Polish space.
A semigroup (𝖯_t)_t≥ 0 of positive contraction operators on C_0(𝒳) is called a Feller (or Feller-Dynkin) semigroup if: (i) 𝖯_t C_0(𝒳)⊂ C_0(𝒳), for all t≥ 0, (ii) 𝖯_tf(x)→ f(x), as t→ 0, for all f∈ C_0(𝒳), x∈𝒳. Associated to (𝖯_t)_t≥ 0 we have a unique Markov process, called a Feller process, on 𝒳.


Now, for each N∈ℕ, let (𝐗^(N)(t);t≥ 0) be a Feller process on ℰ_N with  associated semigroup (𝖯_N(t))_t≥ 0, and suppose these semigroups are consistent 
with (ℒ_N^N+1)_N=1^∞, namely, 

    𝖯_N+1(t)ℒ_N^N+1 = ℒ_N^N+1𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

The following result from <cit.>, essentially Proposition 2.4 therein, gives the existence of a Feller process 𝐗^(∞) on ℰ_∞ consistent with the 𝐗^(N). Because the main idea is beautifully simple and since there are some subtleties with measurability we give some details.


	Let ℰ_∞ be a Feller boundary of (ℰ_N,ℒ_N^N+1)_N=1^∞.  For all N∈ℕ, let (𝖯_N(t))_t≥ 0 be Feller semigroups satisfying (<ref>). Then, there exists a unique Feller Markov semigroup
	(𝖯_∞(t))_t≥ 0 on ℰ_∞, with associated process (𝐗^(∞)(t);t≥ 0), such that 
	
    𝖯_∞(t)ℒ_N^∞
    		= ℒ_N^∞𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

 


To construct the semigroup (𝖯_∞(t))_t≥ 0 we first need to define, for each t ≥ 0, x∈ℰ_∞, a probability measure 𝖯(t)(x,·) on ℰ_∞. Note that, this probability measure needs to satisfy,

    𝖯(t)(x,·)ℒ_N^∞(dy)=ℒ_N^∞𝖯_N(t)(x,dy).

Then, note that, because of (<ref>), and since ℒ_N^∞=ℒ^∞_N+1ℒ_N^N+1, we have

    (ℒ_N^∞𝖯_N(t)(x,·))_N=1^∞∈lim_←Topℳ_p(ℰ_N),

and by virtue of the fact that the map (<ref>) is a Borel isomorphism this defines a unique probability measure 𝖯(t)(x,·) on ℰ_∞ as desired. Observe also that,  since (<ref>) holds for each x∈ℰ_∞, it is equivalent to (<ref>). To see that 𝖯(t):ℰ_∞→ℰ_∞ is a genuine Markov kernel we first observe that the map ℰ_∞→ℳ_p(ℰ_∞) given by x↦𝖯(t)(x,·) is measurable. This is a consequence of the following facts: the map ℰ_∞→ℳ_p(ℰ_∞) which takes x to its delta measure is continuous, the map (<ref>) is a Borel isomorphism and so bi-measurable and the fact that, for each t≥ 0, the map,

    lim_←Topℳ_p(ℰ_N)    →lim_←Topℳ_p(ℰ_N) 
    
    (μ_N)_N=1^∞   ↦(μ_N𝖯_N(t))_N=1^∞,

is continuous, since for each N ∈ℕ, (𝖯_N(t))_t≥ 0 is Feller. Hence, to conclude that, for 𝒜 measurable, x↦𝖯(t)(x,𝒜) is a measurable function, it suffices to observe that the evaluation maps (for measurable 𝒜) ℳ_p(ℰ_∞) → [0,1] given  by μ↦μ(𝒜) are measurable; recall the σ-algebra generated by these maps is the same as the Borel σ-algebra (but the induced topology on the set ℳ_p(ℰ_∞) is different from the weak topology). The semigroup property then follows by virtue of (<ref>) and the semigroup property of the (𝖯_N(t))_t≥ 0. Finally, the Feller continuity property, boils down, by virtue of Lemma <ref>, to the Feller property of the ℒ_N^∞ and (𝖯_N(t))_t≥ 0. See <cit.> for more details.


Note that, the way we obtained (𝖯_∞(t))_t≥ 0 is non-constructive. Thus a direct description of its dynamics becomes a whole separate task. On the other hand, we have obtained a-apriori the highly non-trivial Feller continuity property, which especially when ℰ_∞ turns out to be infinite-dimensional is a major achievement. We also remark that it is easy to see that we can  obtain invariant measures for (𝖯_∞(t))_t≥ 0 from invariant measures for the (𝖯_N(t))_t≥ 0, see <cit.>. Since we will prove a stronger statement in Theorem <ref> below we do not make this explicit. We now move to some important definitions.



 Let 𝒳 be a locally compact Polish space and let (𝖯_t)_t≥ 0 be a Feller semigroup on C_0(𝒳). Its infinitesimal generator 𝒜 is defined to be,
 
    𝒜f=lim_t↓ 0t^-1(𝖯_t f-f),

 whenever the limit exists in C_0(𝒳).
 The domain of 𝒜, that we denote throughout the paper by 𝒟(𝒜), is the set of f∈ C_0(𝒳) for which the limit exists.



In the setting of Theorem <ref>, we denote by 
𝖫_N and 𝖫_∞ the generators, with domains 𝒟(𝖫_N)⊂ C_0(ℰ_N) and 𝒟(𝖫_∞)⊂ C_0(ℰ_∞), associated to the Feller processes (𝐗^(N)(t);t≥ 0) and (𝐗^(∞)(t);t≥ 0) respectively. The following standard definition will be useful.




    
    Let 𝒜 be a Feller generator with domain 𝒟(𝒜). 
A subspace 𝒞⊆𝒟(𝒜) is
    called a core of 𝒜 if the closure of 𝒜|_𝒞 coincides with 𝒜, where, by 𝒜|_𝒞 we mean restriction of 𝒜 to 𝒞.
    
    The generator is clearly uniquely determined by its action on a core. Then, the following very useful lemma, see Proposition 5.2 of <cit.> for a proof, provides a core for 𝖫_∞.

    
    
    In the setting of Theorem <ref>, for each K∈ℕ suppose 𝒞_K⊆𝒟(𝖫_K) is a core for the generator 𝖫_K. The subspace 𝒞_∞⊆ C_0(ℰ_∞) spanned by all subspaces of the form ℒ_K^∞𝒞_K,  K = 1, 2, …, namely

    𝒞_∞def=span(⋃_K=1^∞ℒ_K^∞𝒞_K)

is then a core for the generator 𝖫_∞.




 §.§ Method of intertwiners: Convergence to equilibrium



We now extend the formalism of the method of intertwiners to also deal with convergence to equilibrium for the process on the boundary. The following result, despite its simplicity, appears to be new. Let C_b(𝒳) denote the space of (real-valued) bounded continuous functions on a locally compact Polish space 𝒳.


 
     In the setting of Theorem <ref>, suppose (μ_N)_N=1^∞∈lim_←Topℳ_p(ℰ_N) with  corresponding boundary measure μ_∞∈ℳ_p(ℰ_∞) uniquely determined by,
     
    μ_∞ℒ_N^∞=μ_N,   ∀ N∈ℕ.

     Moreover, suppose that for all (ν_N)_N=1^∞∈lim_←Topℳ_p(ℰ_N), we have
       
    ν_N𝖯_N(t)(f)t→∞⟶μ_N(f),  ∀ f∈ C_b(ℰ_N),  ∀ N ∈ℕ.
  
       Then, for any ν_∞∈ℳ_p(ℰ_∞), we have 
       
    ν_∞𝖯_∞(t)(f)t→∞⟶μ_∞(f),  ∀ f∈ C_b(ℰ_∞).
 
       In particular, μ_∞ is the unique invariant measure of (𝖯_∞(t))_t≥ 0.
    
    
    We show (<ref>) holds for all functions in
𝒞=⋃_N=1^∞ℒ_N^∞C_0(ℰ_N). Observe that, this implies (<ref>) for all f∈ C_0(ℰ_∞) as, by virtue of Lemma <ref>, 𝒞 is a dense subset of C_0(ℰ_∞). This is vague convergence, which together with the fact that the limiting measure μ_∞ is a probability measure, by a standard argument upgrades it to convergence in distribution which proves the theorem.
        
        Fix N∈ℕ and let f_N∈ C_0(ℰ_N). We have using the intertwining relation (<ref>) that
        
    ν_∞𝖯_∞(t)ℒ_N^∞(f_N)=ν_∞ℒ_N^∞𝖯_N(t)(f_N),  ∀ t≥ 0.

        Now, set ν_N=ν_∞ℒ_N^∞. Since (ν_N)_N=1^∞∈lim_←Topℳ_p(ℰ_N), by our assumption ν_N𝖯_N(t)(f_N)→μ_N(f_N) as t→∞. Hence, 
       
    ν_∞ℒ_N^∞𝖯_N(t)(f_N)→μ_N(f_N),   t→∞.

       Recall that μ_N(f_N)=μ_∞(ℒ_N^∞f_N), and thus, display  (<ref>) follows. Then, the fact that μ_∞ is the unique invariant measure follows by a standard argument.
    




 §.§ Projective system from unitarily invariant Hermitian matrices


From now on we focus on two specific projective systems that are related to random matrices. They form the basic integrable structure behind our problem. They come from the study of unitarily invariant measures on infinite Hermitian matrices, see <cit.> for background. We will need several definitions and preliminary results. For the most part, the material in this subsection is standard and has appeared in equivalent forms in <cit.>.

For any N∈ℕ, let 𝕎_N be the Weyl chamber,

    𝕎_N = {𝐱=(x_1,x_2,…,x_N)∈ℝ^N: x_1≥ x_2 ≥…≥ x_N},

The interior of 𝕎_N and 𝕎_N,+ from (<ref>) will be denoted by 𝕎_N^∘ and 𝕎_N,+^∘ respectively.

 For each N∈ℕ, let 
 ℍ(N) denote the space of N× N Hermitian matrices, and ℍ_+(N)⊂ℍ(N) the subspace of non-negative definite ones.
 
 
 We define the eigenvalue maps
 
    𝖾𝗏𝖺𝗅_N : ℍ(N)    →𝕎_N,
    𝐀   ↦𝐱=(x_1≥ x_2 ≥…≥ x_N),

 where 𝐱=(x_1≥ x_2 ≥…≥ x_N) are the ordered eigenvalues of the matrix 𝐀∈ℍ(N). Observe that, 𝖾𝗏𝖺𝗅_N restricts to a map from ℍ_+(N) to 𝕎_N,+.
 
 
 
 We now let, for each N ∈ℕ, 

    Π_N^N+1:ℍ(N+1)   →ℍ(N),
    [𝐀_ij]_i,j=1^N+1   ↦[𝐀_ij]_i,j=1^N,

be the so-called corners map. Also, observe that, Π_N^N+1 restricts to a map from ℍ_+(N+1) to ℍ_+(N). Let 𝕌(N) be the space of N× N unitary matrices. We will denote by 𝐀^† the complex conjugate of a matrix 𝐀. The following definition is one of our basic building blocks.



For any N∈ℕ, we define

    Λ_N^N+1(𝐱,·)def=𝖫𝖺𝗐[𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐔𝖽𝗂𝖺𝗀(𝐱)𝐔^†))],  ∀𝐱∈𝕎_N+1,
   
where 𝐔 is Haar-distributed on 𝕌(N) and 𝖽𝗂𝖺𝗀(𝐱) indicates the diagonal matrix with entries 𝐱.  More generally, we define the composition
 	
    Λ_K^Ndef=Λ_N-1^NΛ_N-2^N-1⋯Λ_K^K+1,  ∀ N>K≥ 1.
 





We write Δ_N for the Vandermonde determinant given by, for 𝐱∈ℝ^N,  
 
    Δ_N(𝐱)=∏_1≤ i<j≤ N (x_i-x_j).

When 𝐱∈𝕎_N+1 and 𝐲∈𝕎_N, 𝐲≺𝐱 denotes interlacing:
	x_1≥ y_1≥…≥ y_N≥ x_N+1. 
It is a classical result by Baryshnikov <cit.> 
 (in fact the computation is implicit in <cit.>, see also <cit.> for an alternative proof) that the following holds. This explicit formula will be useful in the sequel.


 For N∈ℕ, we have
	
    Λ_N^N+1(𝐱,d𝐲)=N!Δ_N(𝐲)/Δ_N+1(𝐱)1_{𝐲≺𝐱}d𝐲,  ∀𝐱∈𝕎_N+1^∘.
 


As expected these Markov kernels are Feller.

 
      Let N>K≥ 1. The Markov kernel Λ_K^N is Feller, that is, for all f∈ C_0(𝕎_K), we have Λ_K^Nf∈ C_0(𝕎_N). Similarly, for all f∈ C_0(𝕎_K,+), we have Λ_K^Nf∈ C_0(𝕎_N,+).
 
 
     The statement follows from Lemma 2.5 of <cit.>, and using the fact that the composition of Feller kernels is Feller. 
 

We now define the infinite dimensional space Ω, an analogue of the space Ω_+ from the introduction, that will allow in the sequel to also consider negative coordinates.


We define the space Ω as follows:

    Ωdef={ω=(𝐱^+,𝐱^-,γ,δ)∈𝕎_∞,+×𝕎_∞,+×ℝ×ℝ_+:
    		∑_i=1^∞(x_i^+)^2+∑_i=1^∞(x_i^-)^2≤δ},
 
endowed with the topology of coordinate-wise convergence. 

 Observe that, Ω like Ω_+ is locally compact, metrizable and separable. Its topology can be metrized with  the following metric 𝖽_Ω, with ω=(𝐱^+,𝐱^-,γ,δ)

    𝖽_Ω(ω,ω̃)=∑_i=1^∞|x_i^+-x̃_i^+|/2^i(1+|x_i^+-x̃_i^+|)+∑_i=1^∞|x_i^–x̃_i^-|/2^i(1+|x_i^–x̃_i^-|)+|γ-γ̃|+|δ-δ̃|.

We note moreover that Ω_+ can be embedded into Ω as follows:

    (𝐱,γ)↦(𝐱,0,γ,∑_j=1^∞ x_j^2),

and observe that that this is a homeomorphism. This embedding is the natural embedding of Ω_+ into Ω and this will be clearer in the sequel.

We now embed 𝕎_N and 𝕎_N,+ into Ω and Ω_+ respectively, in a natural way (again the fact that this is a reasonable thing to do will be clear in what follows).




For each N∈ℕ, we embed the space 𝕎_N into Ω as follows, with 𝐱^(N)∈𝕎_N,
 
    𝐱^(N)↦((N^-1max{x_i^(N),0})_i=1^∞,(N^-1max{-x_N+1-i^(N),0})_i=1^∞,N^-1∑_i=1^Nx_i^(N) ,N^-2∑_i=1^N (x_i^(N))^2),

 where by convention x_i^(N)≡ 0 whenever i∉ 1, N. We denote this embedded point by ω(𝐱^(N)).
Similarly, we can define an embedding of 𝕎_N,+ into Ω_+ as follows, with 𝐱^(N)∈𝕎_N,+,

    𝐱^(N)↦((N^-1x_i^(N))_i=1^∞,N^-1∑_i=1^N x_i^(N)).

Abusing notation, we denote this embedded point in Ω_+ by ω(𝐱^(N)) again.

 
Note that, the embedding of 𝕎_N,+ into Ω_+ matches the embedding of the stochastic dynamics from the introduction.

Now, let 
 ℍ(∞) denotes the space of infinite Hermitian matrices, and write ℍ_+(∞) for the space of non-negative definite infinite Hermitian matrices (defined as the projective limits of the spaces (ℍ(N))_N=1^∞ and (ℍ_+(N))_N=1^∞ respectively under the corners maps Π_N^N+1; we do not make this more precise as we will only need to work with their finite-dimensional projections). Define the map,

    Π_N^∞:ℍ(∞)   →ℍ(N),
    [𝐀_ij]_i,j=1^∞   ↦[𝐀_ij]_i,j=1^N.

Observe that this restricts to a map from ℍ_+(∞) to ℍ_+(N). 

The following definition is important.


 
 
For ω∈Ω, define the infinite Hermitian random matrix  
 
    𝐀_ω^∞def=γ𝐈 + √(δ-∑_i=1^∞[(x_i^+)^2+(x_i^-)^2])𝐆+∑_j=1^∞x_j^+( ξ(j)ξ(j)^†-𝐈) - ∑_j=1^∞x_j^-(ζ(j)ζ^†(j)-𝐈),

where, 𝐆 is an infinite Gaussian Hermitian matrix, namely, 

    𝐆_iid=𝒩(0,1),  ∀ i∈ℕ,  ( 𝐆_ij), ( 𝐆_ij)d=𝒩(0,1/2),  ∀ i<j,

with all independent entries subject to the Hermitian constraint and where 𝒩(0,σ^2) denotes the centered Gaussian random variable of variance σ^2. Moreover, for any j∈ℕ
the ξ(j) and ζ(j) are infinite vectors of i.i.d. (independent, identically distributed) 
 standard complex Gaussian random variables (real and imaginary parts are independent normally distributed random variables with mean zero and variance 1/2).

For ω∈Ω_+, we defined, abusing notation,

    𝐀_ω^∞def=(γ-∑_j=1^∞x_j)𝐈 +∑_j=1^∞x_jξ(j)ξ^†(j),

 where, as before, (ξ(j))_j=1^∞ is a sequence of infinite vectors of i.i.d. complex Gaussian random variables.
 

 We note that the definition of the matrices 𝐀_ω^∞, for ω∈Ω_+, is consistent with viewing ω∈Ω under the embedding (<ref>). We also note that we will never really need to consider the 𝐀_ω^∞ as infinite matrices but rather only consider them through their K× K top-left corner projections Π_K^∞(𝐀_ω^∞). We can now define a family of Markov kernels that play a key role in our argument.


For any K∈ℕ, we define, with (abusing notation) either ω∈Ω or ω∈Ω_+,
  
    Λ_K^∞(ω,·)def=𝖫𝖺𝗐[𝖾𝗏𝖺𝗅_K(Π_K^∞(𝐀_ω^∞))].

    


The following is easy to see.


For any K∈ℕ and ω∈Ω or ω∈Ω_+, Λ_K^∞(ω,·) is a probability measure on 𝕎_K or 𝕎_K,+ respectively. Moreover, we have
    
    Λ_N^∞Λ_K^N = Λ_K^∞,  ∀ N>K.



This first statement is immediate from the definition and the second statement follows by unitary-invariance by virtue of Baryshnikov's result <cit.>.





The kernels Λ_K^∞ are Markov and in fact Feller as expected.

 
    Let K∈ℕ. Then, we have
    
    Λ_K^∞f   ∈ C_0(Ω),    ∀ f∈ C_0(𝕎_K),
    Λ_K^∞f   ∈ C_0(Ω_+),  ∀ f∈ C_0(𝕎_K,+).
 
In particular, the kernels are Feller Markov.
 
 Continuity of ω↦Λ_N^∞(ω,d𝐲), in the sense of weak convergence, was proven in <cit.> (see the proof of Proposition 3.5 therein). From this we also obtain the measurability of ω↦Λ_K^∞(ω,𝒜) for measurable 𝒜. It remains to prove the vanishing at infinity property. In the case of Ω this was shown in <cit.>. We give a more direct argument in the case of Ω_+ (an adaptation will work for Ω as well). Let f∈ C_0(𝕎_K,+) be arbitrary. 
We need to show that for any sequence (ω^(n))_n=1^∞ in Ω_+, with ω^(n)=(𝐱^(n),γ^(n)), such that ω^(n)n→∞⟶∞ (which means γ^(n)→∞), we have (Λ_K^∞f)(ω^(n))n→∞⟶ 0.
To do so, it suffices to show that when γ^(n)n→∞⟶∞, the largest eigenvalue of Π_K^∞(𝐀_ω^(n)^∞) converges to infinity in distribution. Since the eigenvalues of sub-matrices of different orders interlace (Cauchy interlacing theorem), we only need to prove the K=1 case. Note that, from the explicit formula (<ref>), the corresponding random variable 𝗑_n is simply
 
    𝗑_ndef=γ^(n)-∑_j=1^∞x_j^(n) + ∑_j=1^∞Γ_x_j^(n),

 where each Γ_θ denotes a gamma random variable with Laplace transform z↦ (1+θ z)^-1 and they are all independent. Thus, we want to show that 
 𝗑_nn→∞⟶∞
 when γ^(n)n→∞⟶∞. We have two possibilities; either ∑_j=1^∞x_j^(n) remains bounded, in which case the result follows immediately, otherwise, ∑_j=1^∞x_j^(n)n→∞⟶∞  which implies, ∑_j=1^∞Γ_x_j^(n)⟶∞ in distribution. This last claim can easily be seen by virtue of the fact that, for any z,R≥ 0, we have,
 
    ℙ(∑_j=1^∞Γ_x_j^(n)≤ R)   ≤𝔼[exp(-z∑_j=1^∞Γ_x_j^(n))] e^zR≤e^zR/1+z∑_j=1^∞ x_j^(n).

 This completes the proof. 



The following fundamental result is due to Pickrell <cit.> and Olshanski and Vershik <cit.>, in equivalent form (modulo the Feller property of the kernels). In rather less obvious equivalent form, it can be traced back to the work of Schoenberg <cit.> on totally positive functions, see <cit.>. It was translated to the setting of projective systems in <cit.>. A proof, different from <cit.>, directly in this setup and a generalisation is given in <cit.>.


The space Ω is a Feller boundary of (𝕎_N,Λ_N^N+1)_N=1^∞.


The following result is again due to Olshanski and Vershik in equivalent form (modulo the Feller property of the kernels), see Remark 2.11 in <cit.>, .


 The space Ω_+ is a Feller boundary of (𝕎_N,+,Λ_N^N+1)_N=1^∞.


The uniform approximation theorem that we state below, could also be used to prove the results above, by following the argument in Section 3 of <cit.>, modulo an independent proof (we cannot use Lemma <ref> as the argument would be circular) of the density of ∪_N Λ_N^∞ C_0(𝕎_N) and ∪_N Λ_N^∞ C_0(𝕎_N,+) in C_0(Ω) and C_0(Ω_+) respectively. In the combinatorial setting of the Gelfand-Tsetlin graph this is not too difficult, see Section 2.7 in <cit.>, but in the continuous setting there are additional subtleties. We will not pursue this here.




 §.§ Uniform Approximation theorem
 



 
Let (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞ or (𝕎_N,+)_N=1^∞. Let C_c^∞
(ℝ^K) denote the space of (real-valued) smooth and compactly supported functions on ℝ^K. It will be very convenient in what follows to consider the symmetric extension of the kernels given in (<ref>) and (<ref>) to ℝ^K, which we denote by the same notation. Thus, both Λ_K^N(𝐱,d𝐲) and Λ_K^∞(ω,d𝐲) can be viewed as positive measures on ℝ^K with total mass K! (we choose not to normalise them to be probability measures as it will be more convenient) and can act on functions defined on ℝ^K (this will be clear from context). The following is the main technical result of this part of the paper.
    
 Let K∈ℕ.
For all g∈ C_c^∞(ℝ^K),
we have 

    sup_𝐱^(N)∈ℰ_N|Λ_K^∞g(ω(𝐱^(N)))-Λ_K^Ng(𝐱^(N))|→ 0,
      as  N→∞.


The proof of this theorem will be given in Subsection <ref>. The following function spaces will appear frequently below.

Define C_c,sym^∞(ℝ^K) to be the space of smooth compactly supported functions on ℝ^K that are moreover symmetric in their variables. Then, we define 
C_c,sym^∞(ℰ_K) to be the space of functions on ℰ_K which are restrictions of functions C_c,sym^∞(ℝ^K) to ℰ_K.  


Observe that, by definition
 
    ∫_ℰ_Kg(𝐲)Λ_K^N(𝐱,d𝐲)   =
         1/K!∫_ℝ^Kg(𝐲)Λ_K^N(𝐱,d𝐲),
    ∫_ℰ_Kg(𝐲)Λ_K^∞(ω,d𝐲)   =
         1/K!∫_ℝ^Kg(𝐲)Λ_K^∞(ω,d𝐲),   ∀ g∈ C_c,sym^∞(ℝ^K).

 We have the following corollary of Theorem <ref>.
 
Let K ∈ℕ. For all g ∈ C_c,sym^∞(ℰ_K), we have

    sup_𝐱^(N)∈ℰ_N|∫_ℰ_Kg(𝐲)Λ_K^∞(ω(𝐱^(N)),d𝐲)-∫_ℰ_Kg(𝐲)Λ_K^N(𝐱^(N),d𝐲)|→ 0,
      as  N→∞.

 




 §.§ Approximation of Markov processes


In this subsection we show how using the uniform approximation theorem above and an underlying intertwining we can prove a path-space convergence result to the abstract Feller process associated to (𝖯_∞(t))_t≥ 0. 

To be precise, the running assumption is the following. In the setting of Subsection <ref>, let (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞ or (𝕎_N,+)_N=1^∞ and 
(ℒ_N^N+1)_N=1^∞=(Λ_N^N+1)_N=1^∞. Namely, we consider the systems (𝕎_N,Λ_N^N+1)_N=1^∞ and (𝕎_N,+,Λ_N^N+1)_N=1^∞, 
with Λ_N^N+1 given by (<ref>).  As we have seen in Theorems <ref> and <ref>, the Feller boundaries of these systems are given by Ω and Ω_+ respectively. 



We moreover consider, for each N∈ℕ, a Feller Markov process (𝔵^(N)(t);t≥ 0), with continuous sample paths,
on ℰ_N with transition semigroup (𝖯_N(t))_N=1^∞ and we assume that these semigroups are intertwined:

    𝖯_N+1(t)Λ_N^N+1 = Λ_N^N+1𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

Let (𝐗_t^Ω;t≥ 0) and
(𝐗_t^Ω_+;t≥ 0)
be the corresponding Feller processes on Ω and Ω_+ constructed as in Theorem <ref>,  respectively. We denote the Feller semigroup of the process on the boundary, abusing notation, for either Ω or Ω_+, by (𝖯_∞(t))_t≥ 0.
Moreover, recall that we have 

    𝖯_∞(t)Λ_N^∞
    		= Λ_N^∞𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

 
We need two abstract definitions that we will specialise to our setting shortly. 


    Suppose (ℒ_n)_n=1^∞ is a sequence of Banach spaces, with the norm of ℒ_n denoted by ·_n. Let also ℒ be another Banach space with norm denoted by ·. 
    We say that a sequence of vectors (𝐟_n)_n=1^∞ with 𝐟_n∈ℒ_n approximates a vector 𝐟∈ℒ, and we write 𝐟_n→𝐟, if and only if there exist bounded linear operators π_n: ℒ→ℒ_n, satisfying sup_n∈ℕ|π_n|_n<∞, where |·|_n denotes the operator norm of linear operators from ℒ to ℒ_n, such that,
    
    lim_n→∞𝐟_n-π_n 𝐟_n=0.





Let 𝒜_N and 𝒜_∞ be the generators (defined in an analogous way to Definition <ref>) corresponding to strongly continuous semigroups of contractions on the Banach spaces ℒ_N and ℒ_∞, respectively. We say that 𝒜_∞ is approximated by (𝒜_N)_N=1^∞
  if for any 𝐟 in a core (defined in an analogous way to Definition <ref>) of 𝒜_∞, there exist a sequence  𝐟_N∈𝒟(𝒜_N) such that 𝐟_N→𝐟 and 𝒜_N𝐟_N →𝒜_∞𝐟.





The following proposition is key for our purposes. Here, we take the Banach spaces to be ℒ_N=C_0(ℰ_N) and ℒ=C_0(ℰ_∞) and the implicit maps π_N: C_0(ℰ_∞) → C_0(ℰ_N) to be the ones induced by embeddings of ℰ_N into ℰ_∞, namely (<ref>) or (<ref>).



In the setting described above, we let 𝖫_N and 𝖫_∞ denote the generators of the Feller semigroups  (𝖯_N(t))_t≥ 0 and (𝖯_∞(t))_t≥ 0, respectively. Let 𝒞_N=C_c,sym^∞(ℰ_N) and 𝒞_∞ be as in (<ref>). Moreover, assume that, for all N ∈ℕ, 𝒞_N⊆𝒟(𝖫_N) and 𝖫_N𝒞_N ⊂𝒞_N. Then, for any 𝐟∈𝒞_∞, there exists a sequence (𝐟_N)_N=1^∞ with 𝐟_N∈𝒞_N 
such that 𝐟_N→𝐟 and 𝖫_N𝐟_N →𝖫_∞𝐟.


An immediate consequence of Proposition <ref> and Lemma <ref> is the following result.

 
    In the setting of Proposition <ref>, suppose moreover, that for each N∈ℕ,
    𝒞_N
    is a core for 𝖫_N. Then, 𝖫_∞ is approximated by  𝖫_N in the sense of Definition <ref>.


The two main ingredients to prove Proposition <ref> is
Theorem <ref>, in the form of Corollary <ref>, and the intertwining relation (<ref>). Before giving the proof we obtain as an immediate consequence the convergence of the corresponding Markov dynamics on path space.



In the setting described above, let ℰ_N=𝕎_N and 
(𝐗^(N)_t; t≥ 0) be the process on Ω corresponding to the Feller process (𝔵^(N)(t); t≥ 0), having continuous trajectories, with transition semigroup (𝖯_N(t))_t≥ 0,
under the embedding (<ref>) of 𝕎_N into Ω. Suppose moreover that, for each N ∈ℕ, 𝒞_N=C_c,sym^∞(𝕎_N) 
is a core, and invariant under 𝖫_N, the generator of (𝖯_N(t))_t≥ 0. Then, if 𝐗^(N)_0dN→∞⟶𝐗^Ω_0, we have that,

    𝐗^(N)dN→∞⟶𝐗^Ω,   in  C(ℝ_+,Ω).

   
        Using Corollary <ref> (recall that the implicit maps π_N therein are induced by the embeddings (<ref>) of 𝕎_N into Ω), it follows from <cit.>[Chapter 4, Theorem 2.11] that 𝐗^(N)⟶𝐗^Ω, in distribution, in D(ℝ_+,Ω)
        the space of càdlàg functions on ℝ_+ with values in Ω endowed with the Skorokhod topology, see <cit.>. 
       The desired result is then concluded from the well-known fact that the space C(ℝ_+,Ω) is closed in D(ℝ_+,Ω) in the Skorokhod topology and recalling that, for each N∈ℕ, 𝐗^(N)∈ C(ℝ_+,Ω) .
   
   

An analogous statement holds when the processes live in (𝕎_N,+)_N=1^∞. The proof is the same.


 
In the setting described above, let ℰ_N=𝕎_N,+ and 
(𝐗^(N)_t; t≥ 0) be the process on Ω_+ corresponding to the Feller process (𝔵^(N)(t); t≥ 0), having continuous trajectories, with semigroup (𝖯_N(t))_t≥ 0
under the embedding (<ref>) of 𝕎_N,+ into Ω_+. Suppose moreover that, for each N ∈ℕ, 𝒞_N=C_c,sym^∞(𝕎_N,+) 
is a core, and invariant under 𝖫_N, the generator of (𝖯_N(t))_t≥ 0. Then, if 𝐗^(N)_0dN→∞⟶𝐗^Ω_+_0, we have that,

    𝐗^(N)dN→∞⟶𝐗^Ω_+,   in  C(ℝ_+,Ω_+).

  

  Let us write 𝐗_·^Ω_+=((𝗑_i(·))_i=1^∞,γ(·)) below for the process on Ω_+ in coordinates and similarly 𝐗_·^(N)=((𝗑_i^(N)(·))_i=1^∞,γ^(N)(·)) for the embedded process (recall γ^(N)=∑𝗑_i^(N)).
  
  
Observe that, Theorem <ref>, in particular implies that

    𝗑_i^(N)dN→∞⟶𝗑_i,  in  C(ℝ_+,ℝ_+),   ∀ i ∈ℕ,   and ∑_i=1^N𝗑_i^(N)dN→∞⟶γ,  in  C(ℝ_+,ℝ_+).

  

  
  
    In the setting of Theorem <ref>, there exists a coupling of the 𝐗^(N) and 𝐗^Ω_+ on a single probability space such that, almost surely, for any T ≥ 0,
    
    sup_t∈[0,T]∑_i=1^∞(𝗑_i^(N)(t)-𝗑_i(t))^2 N→∞⟶ 0.





     
    We use the Skorokhod representation theorem to couple 𝐗^(N) and 𝐗^Ω_+ on a single probability space, and, abusing notation, denote the new processes by the same symbol. One can then write for all M∈ℕ,
    
    sup_t∈[0,T]∑_i=1^∞(𝗑_i^(N)(t)-𝗑_i(t))^2≤     
    2sup_t∈[0,T]∑_i=M+1^∞(𝗑_i^(N)(t))^2+2sup_t∈[0,T]∑_i=M+1^∞(𝗑_i(t))^2
       +sup_t∈[0,T]∑_i=1^M(𝗑_i^(N)(t)-𝗑_i(t))^2.

    Let ε>0 be given. Observe first that as sup_t∈[0,T]∑_i=1^∞(𝗑_i(t))^2<∞, there exists M_0∈ℕ, such that for all M≥ M_0, 
    
    2sup_t∈[0,T]∑_i=M+1^∞(𝗑_i(t))^2≤ε/4.

    Then, using Theorem <ref> we have 
    
    sup_t∈[0,T]∑_i=1^M_0(𝗑_i^(N)(t)-𝗑_i(t))^2≤ε/4,

    for large enough N. 
    Note moreover that,
    
    sup_t∈[0,T]∑_i=M_0+1^∞(𝗑_i^(N)(t))^2≤sup_t∈[0,T]𝗑_M_0^(N)(t)sup_t∈[0,T]∑_i=M_0^∞𝗑_i^(N)(t).

Now, from (<ref>), we have 
 
    lim_N→∞sup_t∈[0,T]𝗑_M_0^(N)(t)=sup_t∈[0,T]𝗑_M_0(t),  sup_N∈ℕsup_t∈[0,T]∑_i=M_0^∞𝗑_i^(N)(t)def=c<∞.

    Hence, one can pick a possibly larger M_0 such that sup_t∈[0,T]𝗑_M_0(t)<ε/4c also holds, and thus,
    
    lim sup_N ≥ 1sup_t∈[0,T]∑_i=M_0+1^∞(𝗑_i^(N)(t))^2≤ε/4.

    Putting everything together, we conclude (<ref>).
    



    
   When considering processes on 𝕎_N and Ω instead, a very similar argument can be used to show the following convergence in ℓ^3 as a consequence of Theorem <ref>. Below we write 𝐗_·^Ω=((𝗑^+_i(·))_i=1^∞,(𝗑^-_i(·))_i=1^∞,γ(·),δ(·))  for the process on Ω in coordinates and similarly 𝐗_·^(N)=((𝗑^(N),+_i(·))_i=1^∞,(𝗑^(N),-_i(·))_i=1^∞,γ^(N)(·),δ^(N)(·)) for the embedded process.



   
   
    In the setting of Theorem <ref>,  there exists a coupling of the 𝐗^(N) and 𝐗^Ω on a single probability space such that, almost surely, for any T ≥ 0,
    
    sup_t∈[0,T](∑_i=1^∞|𝗑_i^(N),+(t)-𝗑_i^+(t)|^3+∑_i=1^∞|𝗑_i^(N),-(t)-𝗑_i^-(t)|^3)N→∞⟶ 0.



We now prove Proposition <ref>.


   
   We prove the proposition in the case (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞. The exact same argument works for (ℰ_N)_N=1^∞=(𝕎_N,+)_N=1^∞.
   

We then take the Banach spaces (ℒ_N)_N=1^∞=(C_0(𝕎_N))_N=1^∞ and ℒ=C_0(Ω) in the Definition <ref>.
    Considering the definition of 𝒞_∞ from Lemma <ref>,  it only suffices to prove the statement for functions of the form 𝐟=Λ_K^∞g  with some K∈ℕ and g∈𝒞_K.
       We thus fix a positive integer K, and take an arbitrary function g∈𝒞_K. For any N>K, let 
       𝐟_Ndef=Λ_K^Ng and 𝐟def=Λ_K^∞g. Observe that, by the Feller property of the kernels, 
       𝐟_N∈ C_0(𝕎_N) and 𝐟∈ C_0(Ω), respectively. We want to show that 
       
    𝐟_N→𝐟,   as  N→∞,

        in the sense of Definition <ref>. For each N∈ℕ, we then define the operator π_N:C_0(Ω)→ C_0(𝕎_N) as follows
       
    (π_Nf)(𝐱^(N))=f(ω(𝐱^(N))),  𝐱^(N)∈𝕎_N,

       where by ω(𝐱^(N)) we mean the embedding given in (<ref>). It is easy to see  that the operator norm |π_N|_N≤ 1, uniformly in N, and so the conditions given in Definition <ref> are satisfied. We thus need to check that 
       
    π_N𝐟-𝐟_N_N=sup_𝐱^(N)∈𝕎_N|(π_N𝐟)(𝐱^(N))-𝐟_N(𝐱^(N))|N→∞⟶ 0.

       Note that this is exactly the expression in (<ref>) from Corollary <ref>, and so (<ref>) is verified. We next want to show that
              
    𝖫_N𝐟_N→𝖫_∞𝐟,   as  N→∞,

       also holds, that is, 
            
    π_N(𝖫_∞𝐟)-𝖫_N𝐟_N_N→ 0,   as  N→∞.

       
    First, it follows from the intertwining relations (<ref>) and (<ref>) between the semigroups that for all f∈𝒟(𝖫_K), 
    
    𝖫_NΛ_K^Nf   =Λ_K^N𝖫_Kf,  ∀ N>K,
    𝖫_∞Λ_K^∞f   =Λ_K^∞𝖫_Kf.

    In particular, one can check that

    𝐟_N∈𝒟(𝖫_N). We can then write
    
    π_N(𝖫_∞𝐟)-𝖫_N𝐟_N_N=
             π_N(𝖫_∞Λ_K^∞g)-𝖫_NΛ_K^Ng_N
             =π_N(Λ_K^∞𝖫_Kg)-Λ_K^N𝖫_Kg_N.

       Note moreover that, by assumption, the space 𝒞_K is invariant under the action of 𝖫_K  and thus 𝖫_Kg∈𝒞_K. Hence, (<ref>) follows exactly in the same way as (<ref>) from Corollary <ref>. 
   


 §.§ Proof of the uniform approximation theorem


We only consider the case (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞ throughout this section. It is not hard to see, by inspecting (<ref>) and the relevant definitions carefully, that in the case (ℰ_N)_N=1^∞=(𝕎_N,+)_N=1^∞ we are simply taking the supremum over 𝕎_N,+ instead of 𝕎_N while the quantity we are taking the supremum of is exactly the same as for 𝕎_N. Thus, the result for 𝕎_N,+ is a special case of the one over 𝕎_N. We need some preliminaries. 




  §.§.§ Explicit formulae of Markov kernels

We first present an explicit formula for Λ_K^N(𝐱^(N),d𝐲) which is the starting point in our analysis. For any N≥ 2 and 𝐱^(N)∈𝕎_N^∘, we consider the fundamental spline function ℳ(y;𝐱^(N)):ℝ→ℝ, with knots 𝐱^(N), given by 

    ℳ(y;𝐱^(N))= (N-1)∑_i=1^N
            (x_i^(N)-y)_+^N-2/∏_j=1,j≠ i^N(x_i^(N)-x_j^(N)),
 
   where, for z∈ℝ, (z)_+=max{z,0}. One can check that, for all 𝐱^(N)∈𝕎_N^∘, y ↦ℳ(y;𝐱^(N)) is a piecewise polynomial function 
   with N-2 continuous derivatives. Moreover, it satisfies
   
    ∫_ℝℳ(y;𝐱^(N))dy=1.

 
The next theorem, proven by Olshanski in <cit.>, gives an expression for the Markov kernel Λ_K^N defined in (<ref>) in terms of the spline function introduced above.



    Let N≥ 2 and 𝐱^(N)∈𝕎_N^∘. Then, for any 1≤ K≤ N-1, we have

    Λ_K^N(𝐱^(N),d𝐲)   =
    ∏_l=1^K-1N-K+ll[ ℳ(y_K-j+1;x_N-i+1^(N),…,x_K-i+1^(N)) ]_i,j=1^K/∏_j-i≥ N-K+1(x_i^(N)-x_j^(N))Δ_K(𝐲)d𝐲,  ∀𝐲∈𝕎_K.







We note that, the case K=1, boils down to the following identity

    Λ_1^N(𝐱^(N),dy)=ℳ(y;𝐱^(N))dy,  ∀𝐱^(N)∈𝕎_N^∘,

which is an observation first due to Okounkov <cit.>.


The formula from Theorem <ref> is not yet amenable to asymptotic analysis, partly due to the denominator, which may vanish and lead to singularities. Thankfully, we can use a trick based on a formula for derivatives of spline functions to get rid of the denominator entirely. The new formula for Λ_K^N will however involve derivatives of the spline function ℳ which when we extend in the sequel the formula to 𝐱^(N) with coinciding coordinates will need to be interpreted in a weak sense.



In the setting of Theorem <ref>, we have
    
    Λ_K^N(𝐱^(N),d𝐲)=
    (∏_l=1^K-11/l!)
    (ℳ^(K-i)(dy_K-j+1;x_1^(N),…,x_N-i+1^(N)))_i,j=1^KΔ_K(𝐲).

Here, for any j=1,…,K-1,  ℳ^(j) denotes the j-th derivative of the function ℳ with the convention ℳ^(0)=ℳ, 
and, ℳ^(j)(dy,·)=ℳ^(j)(y,·)dy


    We have following relation for the derivative of the spline function in (<ref>), obtained by direct computation,

    ℳ^(1)(y;𝐱^(N))=N-1/x_1^(N)-x_N^(N)(ℳ(y;x_2^(N),…,x_N^(N))-ℳ(y;x_1^(N),…,x_N-1^(N))).

We now consider the formula given in Theorem (<ref>) and repeatedly apply elementary row operations to the determinant in (<ref>) and use (<ref>) to obtain (<ref>).



Let 
g∈  C_c^∞(ℝ^K)
and consider
Λ_K^Ng(𝐱^(N)) for 𝐱^(N)∈𝕎_N^∘. We have
from (<ref>) that

    Λ_K^Ng(𝐱^(N))
       =∫_ℝ^KΛ_K^N(𝐱^(N),d𝐲)g(𝐲)
    
       =
    ∫_ℝ^K(∏_l=1^K-11/l!)
    (ℳ^(K-i)(dy_K-j+1;x_1^(N),…,x_N-i+1^(N)))_i,j=1^KΔ_K(𝐲)g(𝐲).

 Next, one can expand the determinant and use integration by parts for each term in the expansion to rewrite the above expression as 

    Λ_K^Ng(𝐱^(N))   =(-1)^K(K-1)/2∫_ℝ^K∑_σ∈𝕊_Ksgn(σ)∏_j=1^KΛ_1^N-K+j(dy_σ(j);x_1^(N),…,x_N-K+j^(N))
       ×∂_𝐲^σ(Δ_K(𝐲)g(𝐲)),

where 
we used (<ref>) and for σ in 𝕊_K, the symmetric group of degree K, the differential operator ∂^σ acts on functions h∈ C_c^∞(ℝ^K) as follows:

    ∂_𝐲^σh(𝐲)=∂^K(K-1)/2/∂ y_1^σ(1)-1∂ y_2^σ(2)-1⋯∂ y_K^σ(K)-1h(𝐲).

 Now suppose 𝐱∈𝕎_N is arbitrary. Take a sequence (𝐱_n)_n=1^∞ in 𝕎_N^∘ such that 𝐱_nn→∞⟶𝐱. It then follows from Lemma <ref> that both the left and right hand side of equality (<ref>) above, when evaluating at 𝐱_n,  converge to the corresponding terms for 𝐱, as n→∞. This gives an explicit expression of Λ_K^Ng(𝐱) for all g∈ C_c^∞(ℝ^K) and general 𝐱∈𝕎_N.
Note that, this expression matches the one we obtain by formally expanding the following determinant
 
    ∫_ℝ^K(∏_l=1^K-11/l!)
    (ℳ^(K-i)(dy_K-j+1;x_1,…,x_N-i+1))_i,j=1^KΔ_K(𝐲)g(𝐲),

 where, the derivatives correspond to the distributional derivative
 of the kernel ℳ(dy,·). In particular, it is consistent with the formal expansion of the determinant along rows and columns.  From now on, all appearances of these distributional determinants will be understood to be interpreted in the weak sense explained above when applied to C_c^∞(ℝ^K) functions (and the functions we need to apply them to will always be in this space).




We next present an analogous explicit formula for the kernel Λ_K^∞. We do this in the case of Ω. The case of Ω_+ is in fact a special case under the embedding (<ref>). So, suppose ω∈Ω is arbitrary and written in coordinates as ω=(𝐱^+,𝐱^-,γ,δ). We will make very frequent use of the following notation in everything that follows.



 For ω∈Ω, or, abusing notation, for ω∈Ω_+, we define the probability measure ϕ_ω(dy) on ℝ by,

    ϕ_ω(dy)def=Λ_1^∞(ω,dy).



Given arbitrary ω∈Ω as above, we define, for ϵ>0, ω_ϵ∈Ω by:

    ω_ϵ=(𝐱^+,𝐱^-,γ,δ+ϵ).

We observe that, since ϕ_ω_ϵ involves a Gaussian component, it has a smooth density with respect to Lebesgue measure in ℝ, see <cit.>, which we denote by, abusing notation, ϕ_ω_ϵ(y). We write ϕ_ω_ϵ^(ℓ)(dy)=ϕ_ω_ϵ^(ℓ)(y)dy for its ℓ-th derivative. We have the following result.






Let ϵ>0. Then, the Markov kernel Λ_K^∞(ω_ϵ,d𝐲) has an explicit density with respect to Lebesgue measure in 𝕎_K and is given by,

    Λ_K^∞(ω_ϵ,d𝐲)=
              (∏_l=1^K-11/l!)(ϕ_ω_ϵ^(j-1)(dy_i))_i,j=1^KΔ_K(𝐲)d𝐲.
 




This is implicit in the proof of Theorem 7.7 in <cit.>.


Recall (or simply observe) that, as ϵ→ 0, we have the weak convergence

    Λ^∞_K(ω_ϵ,d𝐲)⟶Λ^∞_K(ω,d𝐲).

Now, given g∈ C_c^∞(ℝ^K), we expand the determinant in (<ref>) and use integration  by parts as before and then take ϵ→ 0 by virtue of (<ref>) to make sense of the potentially distributional determinant, for general ω∈Ω,

    (∏_l=1^K-11/l!)(ϕ_ω^(j-1)(dy_i))_i,j=1^KΔ_K(𝐲),

as follows, when acting on g∈ C_c^∞(ℝ^K),
 

    ∫_ℝ^KΛ_K^∞(ω,d𝐲)g(𝐲)
       =(-1)^K(K-1)/2∫_ℝ^K∏_j=1^Kϕ_ω(dy_j)∑_σ∈𝕊_Ksgn(σ)∂_𝐲^σ(Δ_K(𝐲)g(𝐲)),

where the differential operator is the same as in (<ref>).
 
As before, this weak-sense definition is consistent with the formal expansion of the determinant along rows and columns.  From now on, all appearances of these distributional determinants will be understood to be interpreted in the weak sense explained above when applied to C_c^∞(ℝ^K) functions (all functions we need to apply them to will always be in this space).







  §.§.§ Proof of Theorem <ref>

 For any fixed K∈ℕ, N>K, and 𝐱^(N)∈𝕎_N, we consider the following variant of Λ_K^∞,
 
    Λ̃_K^∞(ω(𝐱^(N)), d𝐲)   =
    (∏_l=1^K-11/l!)
    (ϕ_ω(x_1^(N),…,x_N-K+j^(N))^(j-1)(dy_i))_i,j=1^KΔ_K(𝐲).

We note that we only need (and in fact only define) Λ̃_K^∞ for ω∈Ω of the form ω(𝐱^(N)) for 𝐱^(N)∈𝕎_N. Again the determinant in equation (<ref>) above is interpreted in the weak sense when tested against smooth functions of compact support as explained previously. We first prove the desired result with
Λ_K^∞ replaced by
Λ̃_K^∞ in (<ref>).






In the setting of Theorem <ref>, we have
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^Ng(𝐱^(N))|→ 0,  as  N→∞.
 

After establishing Theorem <ref>, we prove that Λ̃_K^∞g(ω(𝐱^(N)))  and Λ_K^∞g(ω(𝐱^(N))) are asymptotically uniformly close as N→∞, see Proposition <ref> for the exact statement, which concludes the desired result.
The proof of Theorem <ref> is by an inductive argument in K. The variant Λ̃_K^∞ of Λ_K^∞ is very well-adapted to proving the rather complicated inductive step and this is its raison d'être.  Before giving the proof, we need  some more preliminaries and notations. 



Denote the Fourier and inverse Fourier transforms of an integrable function f on ℝ^n by f̂ and f̌ respectively given by:

    f̂(ξ)=∫_ℝ^ne^i∑_j=1^ny_jξ_jf(𝐲)d𝐲, f̌(ξ)= (2π)^-n∫_ℝ^ne^-i∑_j=1^ny_jξ_jf(𝐲)d𝐲,  ξ∈ℝ^n.


 We define the following functions
 
    F_+(z;ω)   def=e^iγ z∏_j=1^∞e^-ix_jz/1-ix_jz,  ∀ z∈ℝ,  ω∈Ω_+,
            
    
            F(z;ω)   def=e^iγ z-δ-∑_i=1^∞[(x_i^+)^2+(x_i^-)^2]/2z^2∏_j=1^∞e^-ix_j^+z/1-ix_j^+ze^ix_j^-z/1+ix_j^-z,  ∀ z∈ℝ,  ω∈Ω.

Note that, the infinite products are well-defined by virtue of ω∈Ω_+ and ω∈Ω respectively.



It is easy to check that F_+(·;ω) and F(·;ω) are 
the characteristic functions (i.e. Fourier transforms) of the probability law ϕ_ω when ω∈Ω_+ and Ω, respectively. In particular, we have

    ∫_ℝ F(z;ω)g(z)dz=
         ∫_ℝϕ_ω(dz)ĝ(z),  ∀ g∈𝒮(ℝ),

where 𝒮(ℝ) denotes the space of Schwartz functions on ℝ.  We will also need the following formula in the argument.

    Let 𝐱^(N)∈𝕎_N. Then, we have
    
    ∏_j=1^N(1-izx_j^(N)/N)^-1
        = ∫_-∞^∞(1-izy/N)^-Nℳ(dy;𝐱^(N)).


    
    See Section 8 of <cit.>, stated there for 𝐱^(N)∈𝕎_N^∘ which can be extended to coinciding coordinates by continuity.
    




    Let K=1 and fix g∈ C_c^∞(ℝ). Observe that Λ̃_1^∞=Λ_1^∞. 
    We have by definition that
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_1^∞g(ω(𝐱^(N)))-Λ_1^Ng(𝐱^(N))|   = sup_𝐱^(N)∈𝕎_N|∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)-∫_-∞^∞ℳ(dy;𝐱^(N))g(y)|.

 We have, again by definition,
 
    ∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)
    = 1/2π∫_-∞^∞F(z;ω(𝐱^(N)))(z)ǧ(z)dz.

Using (<ref>) and then (<ref>) we obtain
 
    ∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)   =1/2π∫_-∞^∞∏_j=1^N(1-izx_j^(N)/N)^-1ǧ(z)dz
       = 1/2π∫_-∞^∞∫_-∞^∞(1-izw/N)^-Nℳ(dw;𝐱^(N))
    ǧ(z)dz.

Note now that for any fix w∈ℝ, (1-izw/N)^-N converges pointwise to e^iwz. Also, since
g∈ C_c^∞(ℝ),

the function ǧ is integrable on ℝ. Thus, by dominated convergence theorem,

    𝖦_N(w)   def=1/2π∫_-∞^∞(1-izw/N)^-Nǧ(z)dz-g(w)
    
       =1/2π∫_-∞^∞[(1-izw/N)^-N-e^izw]ǧ(z)dz → 0,   as  N→∞.

 We show in the Lemma <ref> below that the above convergence is uniform over ℝ which proves the result for the case K=1. This is because by the uniform convergence, we have for N large enough that sup_w∈ℝ|𝖦_N(w)|<ε, for a given ε>0,   
 and accordingly,
    
    sup_𝐱^(N)∈𝕎_N|∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)-∫_-∞^∞ℳ(dy;𝐱^(N))g(y)|=sup_𝐱^(N)∈𝕎_N|∫_-∞^∞𝖦_N(w)ℳ(dw;𝐱^(N))|
         =sup_w∈ℝ|𝖦_N(w)|sup_𝐱^(N)∈𝕎_N|∫_-∞^∞ℳ(dw;𝐱^(N))|<ε.

 Here, we used the fact that ∫_-∞^∞ℳ(dw;𝐱^(N))=1 for every 𝐱^(N)∈𝕎_N.





  Let 𝖦_N be the function defined in (<ref>). Then, we have,
  
    sup_w∈ℝ|𝖦_N(w)|N →∞⟶ 0.

  

    Suppose by contradiction that the assertion is not true. Thus, there exist δ>0 and a sequence (w_N_n)_n=1^∞ such that 
    
    |𝖦_N_n(w_N_n)|>δ,  ∀ n∈ℕ.

    There are two possibilities:
    
        
  (i)
        There exists some positive constant L
        such that |w_N_n|≤ L for all n∈ℕ.
        
  (ii)
        The sequence (w_N_n)_n=1^∞ is unbounded.
    
    We show both of these cases lead to a contradiction.
    First we assume (i) occurs. Consider the sequence of functions 
    (𝖦_N_n)_n=1^∞
    on the interval [-L,L]. One can see that this is a sequence of uniformly bounded and 
    
    equicontinuous functions
    on [-L,L], and hence by the Arzelà–Ascoli theorem, there exists a subsequence (𝖦_N_n_k)_k∈ℕ converging uniformly to 0 on [-L,L]. This implies that for large enough k we have
    
    |𝖦_N_n_k(w)|<δ,  ∀ w∈ [-L,L],

    which contradicts (<ref>).
    
    Next, we consider (ii), namely |w_N_n|→∞ as n→∞. 
    Then, there exists a non-decreasing, in absolute value, subsequence of (w_N_n)_n=1^∞  diverging to infinity. We denote this subsequence using the same index, and thus, we have 
    |w_N_1|≤|w_N_2|≤…⟶∞.
    Suppose first that
    lim_N_n→∞|w_N_n|/√(N_n)=0,
    
    and let
    
    f_n(z)def=(
         1-izw_N_n/N_n)^-N_n-e^izw_N_n,    z∈ℝ.

    One can easily check that 
   f_n converges to 0 pointwise,
    and so does 𝖦_N_n(w_N_n).  
    This again violates (<ref>).
    Finally, we assume that 
    
    lim_N_n→∞|w_N_n|/√(N_n)> 0, namely it is equal to a positive constant ℓ, or ∞. 
    Note that in this case, g(w_N_n)=0 for large enough n.
Let 

    ℐ_ndef=∫_-∞^∞(1-izw_N_n/N_n)^-N_nǧ(z)dz,  n∈ℕ,

    We show in the following that ℐ_n→ 0 as n→∞ in both cases.  
    As a result, we conclude in view of (<ref>), that |𝖦_N_n(w_N_n)|< δ for sufficiently large n, which contradicts (<ref>) as well, and completes the proof of the lemma. 
    
    Assume first that lim_n→∞|w_N_n|/√(N_n)= ∞.
     Observe that in this case, 
     (1-izw_N_n/N_n)^-N_nn→∞⟶ 0, 
   and hence, ℐ_nn→∞⟶ 0 (with the same reasoning as before) as desired. Next,  suppose lim_n→∞|w_N_n|/√(N_n)=ℓ>0. Since ǧ is integrable, one can choose M_δ>0 such that

    ∫_ℝ∖ (-M_δ,M_δ)ǧ(z)dz ≤δ/8.

Note also that |(1-izw_N_n/N_n)^-N_n|≤ 1 for all n∈ℕ. So 
we have
 
    |ℐ_n|≤|∫_-M_δ^M_δ(1-izw_N_n/N_n)^-N_nǧ(z)dz|+δ/8.

 Now, since lim_n→∞|w_N_n|/√(N_n)=ℓ, there exists n_0∈ℕ such that |w_N_n|/√(N_n)≤ℓ+1 for all n≥ n_0. We pick such an n_0 so that N_n_0> M_δ^2(ℓ+1)^2 also holds, on account of which we can write a Taylor expansion for the function (1-izw_N_n/N_n)^-N_n on the interval (-M_δ,M_δ). 
 Accordingly, we have for all z∈(-M_δ,M_δ) and n≥ n_0,
 
    (1-izw_N_n/N_n)^-N_n= e^izw_N_n-z^2w_N_n^2/N_n+
        R_n(z),

 where R_n(z)= z^3w_N_n^3/3N_n^2+
    z^4w_N_n^4/4N_n^3+ …=
    O(1/√(N_n)).
 Then, 

    |ℐ_n|   ≤|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_n+R_n(z)ǧ(z)dz|+δ/8
       ≤|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|+|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_n(e^R_n(z)-1)ǧ(z)dz|+δ/8.

 The second integral in the above expression converges to 0 as n→∞, since (e^R_n(z)-1)n→∞⟶ 0 and the integrand is bounded by ǧ up to a multiplicative constant on [-M_δ,M_δ]. Thus, for large enough n∈ℕ, we get  
 
    |ℐ_n|   ≤|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|+2δ/8
    ≤|∫_-∞^∞e^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|+3δ/8.

Note now that for each n∈ℕ, the function 
 e^-z^2w_N_n^2/N_nǧ(z)
 is a Schwartz function and the Fourier transform maps the Schwartz space to itself.
It thus follows that for n large enough, we have

    |∫_-∞^∞e^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|<δ/8,

 and hence,
|ℐ_n|<δ/2. 
This finally concludes the proof of the lemma.
 
  

Suppose (<ref>) holds for K∈ℕ, namely, for all g∈ C_c^∞(ℝ^K), 
we have 
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^Ng(𝐱^(N))|⟶ 0,   as  N→∞.
 
 We need to show that for all g∈ C_c^∞(ℝ^K+1),
 
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|⟶ 0,   as  N→∞.
 
 Let g∈ C_c^∞(ℝ^K+1). We have by definition that 
 
 
    (Λ̃_K+1^∞g)(ω(𝐱^(N)))   =
    (∏_l=1^K1/l!)
    ∫_ℝ^K+1(ϕ_ω(x_1^(N),…,x_N-K-j+1^(N))^(j-1)(dy_i))_i,j=1^K+1Δ_K+1(𝐲)g(𝐲)
    
       = (∏_l=1^K1/l!)
    ∫_ℝ^K+1(ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i,j=1^K+1Δ_K+1(𝐲)g(𝐲).

Also, from (<ref>) we have

    (Λ_K+1^Ng)(𝐱^(N))
    =(∏_l=1^K1/l!)∫_ℝ^K+1(ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i,j=1^K+1Δ_K+1(𝐲)g(𝐲).

Thus, we obtain

    |   Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|
    =
    |∫_ℝ^K+1(∏_l=1^K1/l!)[(ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i,j=1^K+1
       -
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i,j=1^K+1]
    Δ_K+1(𝐲)g(𝐲)|

Now, by expanding both determinants along the first row (recall the definition of the determinant in the weak sense is consistent with expanding along a row or column), we get 

    |   Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|
       =|∫_ℝ^K+1∑_k=1^K+1(∏_l=1^K1/l!)(-1)^1+k[ϕ_ω(𝐱^(N))^(K)(dy_K+2-k)(ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i=2,j=1,j≠ k^K+1
       -ℳ^(K)(dy_K+2-k;𝐱^(N))
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1]Δ_K+1(𝐲)g(𝐲)|
    
       =1/K!|
    ∑_k=1^K+1(-1)^1+k∫_ℝ^K(∏_l=1^K-11/l!)
    [
    (ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i=2,j=1,j≠ k^K+1
       -
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1]
    ∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)
       ×(-1)^K+1-k∫_ℝϕ_ω(𝐱^(N))^(K)(dy_K+2-k)∏_j≠ K+2-k1≤ j≤ K+1(y_K+2-k-y_j)g(𝐲)
    _(-1)^K∫_ℝϕ_ω(𝐱^(N))(dy_K+2-k)h_k(𝐲)
          +∑_k=1^K+1(-1)^1+k∫_ℝ^K(∏_l=1^K1/l!)
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_j≠ ki=2,j=1^K+1∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)
    
       ×(-1)^K+1-k∫_ℝ[ℳ^(K)(dy_K+2-k;𝐱^(N))-ϕ_ω(𝐱^(N))^(K)(dy_K+2-k)]
    ∏_j≠ K+2-k1≤ j≤ K+1(y_K+2-k-y_j)g(𝐲)
    _(-1)^K∫_ℝ[ℳ(dy_K+2-k;𝐱^(N))-ϕ_ω(𝐱^(N))(dy_K+2-k)]h_k(𝐲)|,
 
 where h_k∈ C_c^∞(ℝ^K+1) is defined by
 
    h_k(𝐲)=d^K/dy_K+2-k^K[∏_j≠ K+2-k1≤ j≤ K+1(y_K+2-k-y_j)g(𝐲)].

Thus, we have

    |   Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|≤1/K!∑_k=1^K+1∫_ℝϕ_ω(𝐱^(N))(dy_K+2-k)
    
         |∫_ℝ^K(∏_l=1^K-11/l!)[
    (ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i=2,j=1,j≠ k^K+1
       -
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1]
    ∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)h_k(𝐲)|
    
         +1/K!∑_k=1^K+1(∏_l=1^K1/l!)
    ∫_ℝ^K(ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)
    
       ×|∫_ℝ[ℳ(dy_K+2-k;𝐱^(N))-ϕ_ω(𝐱^(N))(dy_K+2-k)]h_k(𝐲)|.
 
To complete the proof, we need to show that  
 the above expression becomes arbitrarily small, uniformly in 𝐱^(N), for large enough N.
Clearly, it suffices to prove this for each term of the above two series individually. We prove that this is indeed the case for the first series. A completely analogous argument works for the second too.


Let ε>0 be given.
Fix 1≤ k≤ K+1 and consider the k-th term of the first series. For each a∈ℝ, let 𝗁_a∈ C_c^∞(ℝ^K)  be the function defined by

    𝗁_a(𝐲)def=h_k(y_1,…,y_k-1,a,y_k,…,y_K),   𝐲∈ℝ^K.



Since h_k∈ C_c^∞(ℝ^K+1), there exists a compact interval ℑ_k such that 𝗁_a≡ 0 
for all a∉ℑ_k.
Moreover, by virtue of the induction hypothesis, for every a∈ℑ_k, there exist a smallest 𝖭(a)∈ℕ (depending on a), such that for all N≥𝖭(a), we have 

    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞𝗁_a(ω(𝐱^(N)))-Λ_K^N𝗁_a(𝐱^(N))|
    <ε.

Note now that if 𝖬_kdef=sup_a∈ℑ_k𝖭(a) is finite,
then the above inequality holds for all a∈ℑ_k and N≥𝖬_k, and so, the result follows immediately owing to the fact that ϕ_ω(𝐱^(N)) is a probability measure on ℝ for every 𝐱^(N)∈𝕎_N.

We thus wish to show that 𝖬_k<∞. Suppose by contradiction that it is not true. Accordingly, one can find a sequence (𝔞_n)_n=1^∞∈ℑ_k with 𝔞_nn→∞⟶𝔞_*∈ℑ_k such that 
𝖭(n)↗∞.
Let 𝗁_*def=𝗁_𝔞_* and
𝖭_*∈ℕ 
be such that 

    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞𝗁_*(ω(𝐱^(N)))-Λ_K^N𝗁_*(𝐱^(N))|< ε/2
 
for all N≥𝖭_* (note that 𝖭_* is also well-defined by the induction hypothesis).

Now, since h_k∈ C_c^∞(ℝ^K+1) and 𝔞_nn→∞⟶𝔞_*, we have 𝗁_ndef=𝗁_𝔞_nn→∞⟶𝗁_* pointwise. Moreover, 
it follows from the Arzelà–Ascoli theorem that there exists a uniformly convergent sub-sequence of this sequence which we denote by the same notation. 
Hence, we can pick 𝗆∈ℕ large enough so that for all n≥𝗆, we have 𝖭(n)> 𝖭_* and 

    |𝗁_n(𝐲)-𝗁_*(𝐲)|<ε/4,  ∀𝐲∈ℝ^K.
 
Therefore, we obtain 
 
    |Λ̃_K^∞𝗁_𝗆(ω(𝐱^(N)))-Λ_K^N𝗁_𝗆(𝐱^(N))|   ≤|Λ̃_K^∞(𝗁_𝗆-𝗁_*)(ω(𝐱^(N)))|+|Λ_K^N(𝗁_𝗆-𝗁_*)(𝐱^(N))|
         +|Λ̃_K^∞𝗁_*(ω(𝐱^(N)))-Λ_K^N𝗁_*(𝐱^(N))|
       <
         ε/4+ε/4+ε/2=ε,

 for all N≥𝖭_*, and 𝐱^(N)∈𝕎_N. This implies that
 
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞𝗁_𝗆(ω(𝐱^(N)))-Λ_K^N𝗁_𝗆(𝐱^(N))|<ε,  ∀ N≥𝖭_*.

 Thus, we must have 𝖭(𝗆)≤𝖭_* which is a contradiction in view of the particular choice of 𝖭(𝗆) and 𝗆.

We finally conclude Theorem <ref>  by proving the following proposition.



Let K∈ℕ. Then, for all g∈ C_c^∞(ℝ^K) we have:

    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^∞g(ω(𝐱^(N)))|→ 0,
      as  N→∞.





    We have by definition
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^∞g(ω(𝐱^(N)))|
         =sup_𝐱^(N)∈𝕎_N|
    ∫_ℝ^K(∏_l=1^K-11/l!)
    ((ϕ_ω(x_K-j+1^(N),…,x_N^(N))^(j-1)(dy_K+1-i))_i,j=1^K -
    (ϕ_ω(𝐱^(N))^(j-1)(dy_K+1-i))_i,j=1^K)h(𝐲)|,

with 
h(𝐲)=Δ_K(𝐲)g(𝐲).
We show that the above expression is arbitrarily small for large enough N. Define the following K × K matrices,

    𝐅̃(𝐳;𝐱^(N))=
        [   (-iz_K)^K-1∏_j=1^N(1-iz_Kx_j^(N)/N)^-1                                        …   (-iz_1)^K-1∏_j=1^N(1-iz_1x_j^(N)/N)^-1; (-iz_K)^K-2∏_j=2^N(1-iz_Kx_j^(N)/N-1)^-1                                        … (-iz_1)^K-2∏_j=2^N(1-iz_1x_j^(N)/N-1)^-1;                                        ⋮                                        ⋱                                        ⋮;          ∏_j=K^N(1-iz_Kx_j^(N)/N-K+1)^-1                                        …          ∏_j=K^N(1-iz_1x_j^(N)/N-K+1)^-1 ],
    𝐅(𝐳;𝐱^(N))=
        [ (-iz_K)^K-1∏_j=1^N(1-iz_Kx_j^(N)/N)^-1                                      … (-iz_1)^K-1∏_j=1^N(1-iz_1x_j^(N)/N)^-1; (-iz_K)^K-2∏_j=2^N(1-iz_Kx_j^(N)/N)^-1                                      … (-iz_1)^K-2∏_j=2^N(1-iz_1x_j^(N)/N)^-1;                                      ⋮                                      ⋱                                      ⋮;            ∏_j=K^N(1-iz_Kx_j^(N)/N)^-1                                      …            ∏_j=K^N(1-iz_1x_j^(N)/N)^-1 ].

We now use (<ref>) repeatedly to rewrite the determinants in (<ref>) in terms of the determinants of the corresponding Fourier transforms.  
Then, we do elementary row operations repeatedly for the second determinant.
We finally use (<ref>) and expand both determinants in order to obtain

    ∫_ℝ^K   [(ϕ_ω(x_K-j+1^(N),…,x_N^(N))^(j-1)(dy_K+1-i))_i,j=1^K -
    (ϕ_ω(𝐱^(N))^(j-1)(dy_K+1-i))_i,j=1^K]Δ_K(𝐲)g(𝐲)
    
       =(2π)^-K∫_ℝ^K[(𝐅̃(𝐳;𝐱^(N)))
    -(𝐅(𝐳;𝐱^(N)))]
    ȟ(𝐳)d𝐳
       =(2π)^-K∑_σ∈𝕊_Ksgn(σ)∫_ℝ^K∫_ℝ^K[∏_l=1^K(1-iz_σ(l)w_σ(l)/N-l+1)^-(N-l+1)-∏_l=1^K(1-iz_σ(l)w_σ(l)/N)^-(N-l+1)] 
    
       ×(∏_l=1^K(-iz_σ(l))^K-lȟ(𝐳)d𝐳)∏_l=1^Kℳ(dw_σ(l);x_l^(N),…,x_N^(N)).

Note now that for any fixed w_σ(l), the expression in the brackets on the RHS goes to 0 pointwise in z_σ(l), and moreover, it is bounded. Therefore, since the function in the parentheses is integrable, the inner integral converges to 0 by the dominated convergence theorem. One can then use a similar argument as in the proof of Lemma <ref> to verify that the convergence is indeed uniform 
and hence, the expression can be made arbitrarily small by picking large enough N. Finally, the conclusion follows by fact that ℳ(dw;𝐱) is a probability measure on ℝ.  








§ PROPERTIES OF THE FINITE-DIMENSIONAL DIFFUSIONS
  


 §.§ Well-posedness and Feller property


In this subsection we present several, for the most part standard, results on the dynamics (<ref>) that we will need.
 

 
The SDE (<ref>) has a unique strong non-exploding solution (𝔵^(N)(t);t≥ 0) for any initial condition 𝔵^(N)(0)=𝐱∈𝕎_N,+. Moreover, almost surely, for all t>0, 𝔵^(N)(t)∈𝕎_N,+^>, where

    𝕎_N,+^>={𝐱∈ℝ^N:x_1>x_2>⋯ >x_N≥ 0}.

Finally, if 𝔵^(N)(0)=𝐱∈𝕎_N,+^∘, then almost surely for all t≥ 0, 𝔵^(N)(t)∈𝕎_N,+^∘.



All statements, except the last one, follow from the main theorem of <cit.> by verifying the elementary conditions therein. To establish the last statement, observe that since 𝐱∈𝕎_N,+^∘, using Itô's formula we have, for all t≥ 0,

    log𝔵_N^(N)(t)    = log x_N+ 𝗐_N(t) -1+η/2t+∫_0^t1/2𝔵_N^(N)(s)ds+∫_0^t∑_j=1^N-1𝔵_j^(N)(s)/𝔵_N^(N)(s)-𝔵_j^(N)(s)ds
       ≥log x_N+ 𝗐_N(t) -1+η/2t-(N-1)t(inf_s∈[0,t]|1-𝔵_N^(N)(s)/𝔵_N-1^(N)(s)|
    )^-1.

The conclusion then follows since the RHS is finite.



 The Markov semigroup associated with (𝔵^(N)(t);t≥ 0), the unique strong solution of (<ref>), is denoted by (𝔓_N(t))_t≥ 0.




We next introduce certain matrix-valued dynamics (𝐇_t^(N);t≥ 0) from <cit.> whose eigenvalue evolution is described by the SDE (<ref>).
Recall that ℍ_+(N) denotes the space of N× N non-negative definite Hermitian matrices.
We denote by Tr(𝐀) the trace of  a matrix 𝐀. Consider the following ℍ_+(N)-valued SDE introduced by Rider-Valko in <cit.>,

    d𝐇_t^(N)=1/2(dΓ_t^(N)𝐇_t^(N)+𝐇_t^(N)d(
     Γ_t^(N))^†)+(-η+N/2𝐇_t^(N)+1/2(1+Tr(𝐇_t^(N)))𝐈)dt,

where 
   (Γ_t^(N);t ≥ 0) is a matrix-valued complex Brownian motion; namely, an N× N matrix with the entries independent complex Brownian motions (the real and imaginary parts are independent standard real Brownian motions), and
𝐈 is the N× N identity matrix.

For any 𝐇_0^(N)∈ℍ_+(N), the SDE (<ref>) has a unique strong solution.


    It is easy to see that the coefficients of the equation are Lipschitz continuous in  matrix norm, and thus existence and uniqueness of the strong solution follows from well-known results, see for example <cit.>. 


 
Let 𝐱∈𝕎_N,+ and 𝐇∈ℍ_+(N) be such that 𝖾𝗏𝖺𝗅_N(𝐇)=𝐱. 
Let (𝐇_t^(N);t≥ 0) be the solution to (<ref>) starting from 𝐇 and (𝔵^(N)(t);t≥ 0) be the solution to (<ref>) starting from 𝐱. Then,
    
    (𝖾𝗏𝖺𝗅_N(𝐇^(N)_t);t≥ 0)d=(𝔵^(N)(t);t≥ 0).




The statement is a consequence of the general result of <cit.>, which relates matrix diffusions solving SDE generalising (<ref>) to their eigenvalues evolutions given in terms of SDE generalising (<ref>). The computation is also implicit in <cit.>.





By looking at the stochastic equation satisfied by the process t↦(𝐇_t^(N)), see proof of Proposition 12 in 
<cit.>, we can improve the statement of Lemma <ref> regarding never reaching the origin, to initial conditions 𝐱 with coinciding coordinates.




 Let η∈ℝ and N∈ℕ. The semigroup (𝔓_N(t))_t≥ 0 is a Feller semigroup.



    The semigroup associated to (𝐇_t^(N);t≥ 0), as a unique strong solution to a SDE with Lipschitz coefficients, is Feller by classical results, see <cit.>. The Feller property of the eigenvalue process follows from that of the matrix process. More precisely,
    let (𝖰_N(t))_t≥ 0 denote the semigroup associated to (𝐇_t^(N);t ≥ 0). Note that, if 𝐱=𝖾𝗏𝖺𝗅_N(𝐇), one has from Lemma <ref>,
    
    (𝔓_N(t)f)(𝐱)=(𝖰_N(t)f∘𝖾𝗏𝖺𝗅_N)(𝖽𝗂𝖺𝗀(𝐱)),

   and moreover, note that if f ∈ C_0(𝕎_N,+) then f ∘𝖾𝗏𝖺𝗅_N ∈ C_0(ℍ_+(N)). From this, and the Feller property of (𝖰_N(t))_t≥ 0, one immediately checks the Feller property for (𝔓_N(t))_t≥ 0. 



By hypoellipticity <cit.> of the generator of (𝖰_N(t))_t≥ 0, shown in Proposition 12 of <cit.>, we have that 𝖰_N(t)(𝐀,d𝐁)=𝖰_N(t)(𝐀,𝐁)d𝐁, with d𝐁 Lebesgue measure on ℍ_+(N), where the density 𝖰_N(t)(𝐀,𝐁) is smooth, viewed as a function in the matrix entries, see <cit.>.  



Our next goal is to give a description of the N-dimensional dynamics (𝔓_N(t))_t≥ 0, in some sense, in terms of one-dimensional diffusions. For this, we define for fixed N∈ℕ and η∈ℝ, consider unique strong solution to the following SDE in [0,∞),

    d𝗓^(N)(t)    = 𝗓^(N)(t) d𝗐(t) + [(1-η/2-N)𝗓^(N)(t)+1/2]dt,

with 𝗐 as a usual a standard Brownian motion, with infinitesimal generator,

    𝖫_x^(N) = x^2/2∂_x^2+((1-η/2-N)x+1/2)∂_x
 
and transition density with respect to Lebesgue measure in [0,∞) that we denote by 𝔭^(N)_t(x,y). It is known that ∞ is a natural boundary and 0 an entrance boundary in the terminology of Feller <cit.>. In particular, almost surely, for all strictly positive times the diffusion 𝗓^(N) is in (0,∞). 

We then have the following proposition. Analogous results are well-known for certain generalisations of Dyson Brownian motion: the radial Dunkl <cit.> and Heckman-Opdam <cit.> processes.


  Let η∈ℝ and N∈ℕ. Then, C^∞_c,sym(𝕎_N,+) forms a core for the generator of the semigroup (𝔓_N(t))_t≥ 0  and it is moreover invariant under the action of the generator. Finally, the kernel 𝔓_N(t)(𝐱,d𝐲) of 𝔓_N(t) has an explicit density with respect to the Lebesgue measure in 𝕎_N,+ given by (abusing notation), for 𝐱∈𝕎_N,+^∘, t>0,

    𝔓_N(t)(x,y)= e^-λ_NtΔ_N(𝐲)/Δ_N(𝐱)(𝔭_t^(N)(x_i,y_j))_i,j=1^N,

with λ_N=N(N-1)(1-3η/2-2N)/6.
   


 Let us denote by (𝖫_N,𝒟(𝖫_N)) the generator of (𝔓_N(t))_t≥ 0. Define the formal differential operator 𝐋_N by,

    𝐋_N=∑_i=1^N x_i^2/2∂_x_i^2+∑_i=1^N[-η/2x_i+1/2+∑_j=1,j≠ i^Nx_ix_j/x_i-x_j]∂_x_i.

We will identify shortly 𝖫_N with 𝐋_N on C^∞_c,sym(𝕎_N,+).

We first check that C_c,sym^∞(𝕎_N,+) is invariant under 𝐋_N. Let g∈ C_c,sym^∞(𝕎_N,+) be arbitrary and let f be a function in C_c,sym^∞(ℝ^N) that restricts to g∈ C_c,sym^∞(𝕎_N,+). Consider 𝐋_Nf, where we view 𝐋_N as acting on functions on ℝ^N in the obvious way, and observe that on 𝕎_N,+ this restricts to 𝐋_N g. It then suffices to show 𝐋_Nf ∈ C_c,sym^∞(ℝ^N). 𝐋_Nf is clearly of compact support and moreover it is symmetric in its arguments. Showing smoothness boils down to showing that the only part which is possibly singular,

    (x_1,…, x_N)   ↦∑_1≤ i< j≤ N x_i x_j ∂_x_if(x_1,…,x_N)-∂_x_jf(x_1,…,x_N)/x_i-x_j,

is actually smooth on the whole of ℝ^N. This moreover boils down to proving the following claim. For a smooth symmetric function H on ℝ^2, the set of which we denote by C_sym^∞(ℝ^2), the function 

    (x,y) ↦∂_x H(x,y)-∂_yH(x,y)/x-y

is smooth on the whole of ℝ^2, which is a direct consequence of known results. Namely, for H∈ C_sym^∞(ℝ^2) let ℌ be the unique function such that 

    H(x,y)=ℌ(S_1(x,y),S_2(x,y)),

where S_1(x,y)=-(x+y) and S_2(x,y)=xy, see <cit.>.
Then, we have, see <cit.>,

    ∂_x H(x,y)-∂_yH(x,y)/x-y=-∂ℌ/∂S_2.

Finally, from Corollary 3.3 of <cit.>, ℌ is a smooth function and this establishes the claim. 

We now show that 𝖫_N extends 𝐋_N on C_0(𝕎_N,+) and C^∞_c,sym(𝕎_N,+) is a core for it. First, for f so that 𝐋_Nf makes sense, define the process t↦𝖬_f(t) by, 

    𝖬_f(t)def=f(𝗓(t))-f(𝗓(0))-∫_0^t 𝐋_N f(𝗓(s))ds,

where 𝖫𝖺𝗐(𝗓)=𝖫𝖺𝗐(𝔵), with 𝔵 the unique strong solution to (<ref>), namely the process with generator 𝖫_N. Observe that, for f ∈ C^∞_c,sym(𝕎_N,+), 𝖬_f is bounded since 𝐋_N C^∞_c,sym(𝕎_N,+) ⊂ C^∞_c,sym(𝕎_N,+). Hence, an application of  Itô's formula <cit.> to the solution of the SDE (<ref>) gives that t↦𝖬_f(t) is a martingale (with respect to the natural filtration of the coordinate process). In particular, from Theorem 17.23 of <cit.> we obtain that 𝖫_N extends (we do not know yet if this extension is unique) 𝐋_N|_C^∞_c,sym(𝕎_N,+) to C_0(𝕎_N,+).  

It remains to show that C^∞_c,sym(𝕎_N,+) is actually a core for the generator 𝖫_N. This is done by connecting the SDE (<ref>) to a well-posed martingale problem. Define C_𝐱(ℝ_+,𝕎_N,+)={f∈ C(ℝ_+,𝕎_N,+):f(0)=𝐱}.
We say that a family of probability measures {𝐏_𝐱}_𝐱∈𝕎_N,+ on C(ℝ_+,𝕎_N,+), endowed with the filtration generated by the coordinate process, is a solution to the martingale problem for (𝐋_N,C^∞_c,sym(𝕎_N,+)) if 𝐏_𝐱(C_𝐱(ℝ_+,𝕎_N,+))=1, and under 𝐏_𝐱, for every f∈ C^∞_c,sym(𝕎_N,+), the process t↦𝖬_f(t) defined in (<ref>), where 𝖫𝖺𝗐(𝗓)=𝐏_𝐱,
 is a martingale. We now show that this martingale problem is well-posed, namely has a unique solution. First, the law of any weak solution, to the SDE (<ref>) starting from 𝐱, gives rises to a solution to the martingale problem by an application of Itô's formula <cit.>. We then show that this solution to the martingale problem must be unique. Consider an arbitrary solution to the martingale problem. For R>0, define the stopping times,
 
    ζ_R=inf{t≥ 0;||𝗓(t)||≥ R   or ∑_1=i≠ j=N|∫_0^t𝗓_i(s)𝗓_j(s)/𝗓_i(s)-𝗓_j(s)ds |≥ R },

 where, for 𝐱∈ℝ^N, ||𝐱|| is its Euclidean norm. For i,j=1,…,N, we choose f_i,f_ij∈ C_c^∞(𝕎_N,+) satisfying f_i(𝐱)=x_i, f_ij(𝐱)=x_ix_j, for ||𝐱||≤ R. We now claim that, for all i,j=1,…,N, the processes defined by,

    t ↦𝖭_i(t∧ζ_R)def=𝖬_f_i^R(t∧ζ_R)=𝗓_i(t∧ζ_R)-𝗓_i(0)-∫_0^t∧η_R(-η/2𝗓_i(s)+1/2+∑_j=1,j≠ i^N𝗓_i(s) 𝗓_j(s)/𝗓_i(s)-𝗓_j(s))ds

and t ↦𝖭_ij(t∧ζ_R)def=𝖬_f_ij^R(t∧ζ_R) are martingales. We approximate f_i^R and f_ij^R by functions in C_c,sym^∞(𝕎_N,+), for which we know the martingale property of 𝖬_f. By the Stone-Weirstrass theorem the space C_c,sym^∞(𝕎_N,+) is dense in C_0(𝕎_N,+) but we need slightly more in convergence of the derivatives.
More generally, let us take a function g∈ C_c^∞(𝕎_N) (note that we can view the functions f_i^R,f_ij^R as in C_c^∞(𝕎_N)). Then, extend g by symmetry to the whole of ℝ^N to obtain a function 𝖦. Observe that, by construction 𝖦 is symmetric, has compact support, is smooth except on the hyperplanes {x_i_1=x_i_2=⋯=x_i_ℓ}, is Lipschitz on the whole of ℝ^N and 𝖦|_𝕎_N=g. Let 𝔲 be a symmetric approximation of the identity and define, for each m ∈ℕ,

    𝔤_m(𝐱)def=∫_ℝ^N𝖦(𝐲) m^N𝔲(m(𝐱-𝐲))d𝐲.

Then, observe that, 𝔤_m ∈ C^∞_c,sym(ℝ^N) and that 𝔤_m approximates, as m →∞, g in C_0(𝕎_N) and all its derivatives approximate uniformly on compact sets in 𝕎_N^∘ the derivatives of g. From the above we get that there exist sequences (𝔣_i^m)_m=1^∞, (𝔣_ij^m)_m=1^∞ in C_c,sym^∞(𝕎_N,+) so that, as m →∞, for all i,j=1,…,N,

    𝖬_𝔣_i^m(·∧ζ_R) ⟶𝖬_f_i^R(·∧ζ_R),  𝖬_𝔣_ij^m(·∧ζ_R) ⟶𝖬_f_ij^R(·∧ζ_R),

uniformly on compact sets, in distribution under 𝐏_𝐱. Hence, we easily obtain that for all i,j=1,…,N, the processes t ↦𝖬_f_i^R(t ∧ζ_R), t ↦𝖬_f_ij^R(t ∧ζ_R) are martingales as desired. Then, we can follow the computation in Theorem 32.7 of <cit.> or Proposition 5.4.6 of <cit.> to obtain independent standard Brownian motions (𝗐_i(·))_i=1^N (with respect to possibly an extension of the original filtration), such that, for all i=1,…,N, 
 
    𝖭_i(t∧ζ_R)=∫_0^t∧ζ_R𝗓_i(s)d𝗐_i(s),   ∀ t≥ 0.

 In particular, we obtain that (𝗓_i(·∧ζ_R),𝗐_i(·∧ζ_R))_i=1^N is a weak solution to the SDE (<ref>) up until time ζ_R. But by pathwise uniqueness shown in <cit.>, this solution must be equal to the unique strong solution of (<ref>) up until ζ_R. Observe that, under the law of the solution to (<ref>), ζ_R→∞, as R →∞. In particular, ζ_R →∞, as R→∞, under 𝐏_𝐱 and from a solution to the martingale problem we constructed a solution to (<ref>) for all times. Hence, the solution to the martingale problem for (𝐋_N,C^∞_c,sym(𝕎_N,+)) must be unique (for otherwise we would have constructed two different in law solutions to (<ref>)). The associated diffusion process to it, see <cit.>, is clearly nothing but the solution to the SDE (<ref>), namely the Feller process with semigroup (𝔓_N(t))_t≥ 0. By virtue of Theorem 2.5 of <cit.>, since C_c,sym^∞(𝕎_N,+) is dense C_0(𝕎_N,+) and the martingale problem for (𝐋_N,C^∞_c,sym(𝕎_N,+)) is well-posed, we obtain that operator 𝐋_N is closable in C_0(𝕎_N,+) and the closure is nothing but the generator 𝖫_N of the semigroup (𝔓_N(t))_t ≥ 0. Namely 𝖫_N|_C_c,sym^∞(𝕎_N,+)=𝐋_N|_C_c,sym^∞(𝕎_N,+)=𝖫_N or in other words, C_c,sym^∞(𝕎_N,+) is a core for (𝖫_N,𝒟(𝖫_N)).



The final statement in the proposition is a direct consequence of the following observation. The generator 𝐋_N acting on g∈ C_c,sym^∞(𝕎_N,+) can be written, after an elementary computation, as

    𝐋_Ng(x_1,…,x_N)= [Δ_N(𝐱)^-1∘(∑_i=1^N 𝖫_x_i^(N))∘Δ_N(𝐱)-λ_N]g(x_1,…,x_N),

which is exactly the Doob h-transform, see <cit.>, of N independent one-dimensional diffusions with generator 𝖫^(N) killed when they intersect, by the Vandermonde determinant Δ_N. This is the Markov process which has transition kernel having density (<ref>), by virtue of the Karlin-McGregor formula <cit.>, and the conclusion follows. 






 §.§ Consistency


The following consistency or intertwining relation between (𝔓_N(t))_t≥ 0 and (𝔓_N+1(t))_t≥0 is  the main algebraic ingredient for the method of intertwiners.

         We have the intertwining relations:
         
    𝔓_N+1(t)Λ_N^N+1=Λ_N^N+1𝔓_N(t),  ∀ t≥0, ∀ N∈ℕ.

     

We present two arguments for a proof. The first argument is a little more sophisticated and uses matrix stochastic calculus. It is conceptually more natural as it somehow explains the origin of the intertwining: the intertwining (<ref>) simply follows from consistency with respect to corners projection at the level of the matrix diffusions. On the other hand, we only give a full proof for the special initial condition 𝐇^(N)_0=0 which allows for important simplifications (in the case of Dyson Brownian motion which is also consistent with Λ_N^N+1 the adaptation of the argument works for any initial condition).  The second argument is a direct verification, which works for all initial conditions, and thus is a complete proof, using the explicit formula from (<ref>).

We begin with a little formalism giving consistency at the level of the eigenvalues from consistency at the level of the matrix process.

    
        Let (𝐂_t^(N);t≥ 0) and (𝐂_t^(N+1);t≥ 0) be stochastic processes on ℍ(N) and ℍ(N+1) respectively. Suppose that (𝖾𝗏𝖺𝗅_N(𝐂_t^(N));t≥ 0) and (𝖾𝗏𝖺𝗅_N+1(𝐂_t^(N+1));t≥ 0) are Markovian with semigroups (𝖯_N(t))_t≥ 0 and (𝖯_N+1(t))_t≥ 0 respectively. Assume that, for all t≥ 0,
       
    (𝐔^(K))^†𝐂_t^(K)𝐔^(K)d=𝐂_t^(K),  ∀𝐔^(K)∈𝕌(K),   K=N,N+1.

        Suppose moreover that there exists a set 𝒜⊆ℍ(N+1) such that for all 𝐂_0^(N+1)∈𝒜,
        
    Π_N^N+1(𝐂_t^(N+1))d=𝐂_t^(N),   ∀ t ≥ 0.

        Then, we have
        
    𝖯_N+1(t)Λ_N^N+1(𝐱,·)=Λ_N^N+1𝖯_N(t)(𝐱,·),  ∀𝐱∈𝖾𝗏𝖺𝗅_N+1(𝒜),  ∀ t≥ 0.

     
    
    Fix 𝐱 so that 𝐱=𝖾𝗏𝖺𝗅_(N+1)(𝐂_0^(N+1)), with 𝐂_0^(N+1)∈𝒜. We claim that, the LHS of (<ref>) is  given by,
    
    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐂_t^(N+1)))),

     while the RHS by,
     
    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐂_t^(N))).

     These are equal by virtue of (<ref>) and the conclusion follows. To see the claim, first observe that, 
     
    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐂_t^(N)))(d𝐲)=μ𝖯_N(t)(d𝐲),

     where, by unitary invariance (<ref>), μ is given by,
     
    μ(d𝐲)def=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐂_0^(N)))(d𝐲)=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐂_0^(N+1))))(d𝐲)=Λ_N^N+1(𝐱,d𝐲).

     On the other hand, again by unitary invariance (<ref>), we also have, 

    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐂_t^(N+1))))(d𝐲)=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N+1(𝐂_t^(N+1)))Λ_N^N+1(d𝐲)=ν𝖯_N+1(t)Λ_N^N+1(d𝐲),

where ν is given by,

    ν(d𝐲)def=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N+1(𝐂_0^(N+1)))=1_𝐱(d𝐲).

 This completes the proof.
    
    
     We will apply this proposition (with the obvious notational identifications) to prove Theorem <ref> in the special case 𝐇_0^(N)=0, namely for 𝐱=0.
     
     
     Let (𝐇_t^(N); t≥ 0) be the unique strong solution of the equation (<ref>) starting from 𝐇_0^(N)=0, where 0 denotes N× N zero matrix. We have
    
    𝐇_t^(N)d=Π_N^N+1(𝐇_t^(N+1)),  ∀ t≥ 0.



We first  show that the solution to the SDE (<ref>) can be solved explicitly in terms of the matrix exponential Brownian motion (𝐌_t^(θ);t≥ 0), which we introduce next. This is given by the unique strong solution of the matrix SDE: 
    
    d𝐌_t^(θ) =1/2𝐌_t^(θ)d𝐖_t^(N) -θ𝐌_t^(θ)dt,

where θ∈ℝ and (𝐖^(N)_t;t≥ 0) is an N× N complex matrix-valued Brownian motion.  Below we use the notation 𝐀^-† for (𝐀^†)^-1.





The unique strong solution (𝐇_t^(N);t≥ 0) of (<ref>), starting from 𝐇^(N)_0∈ℍ_+(N) is given by

    ((𝐌_t^(-ν_N))^-1(𝐇_0 + 1/2∫_0^t𝐌_s^(-ν_N)(𝐌_s^(-ν_N))^†ds)(𝐌_t^(-ν_N))^-†;t≥ 0).

where ν_N=1/4(η+N), with η being the same parameter as throughout this paper.








First, note that from (<ref>) we have
    
    d(𝐌_t^(θ))^† =1/2d(𝐖_t^(N))^†(𝐌_t^(θ))^† -θ(𝐌_t^(θ))^†dt.

Moreover, using Itô's formula we obtain 
    
    d(𝐌_t^(θ))^-1   = - 1/2d𝐖_t^(N)(𝐌_t^(θ))^-1 +θ(𝐌_t^(θ))^-1dt
    d(𝐌_t^(θ))^-†   = -1/2(𝐌_t^(θ))^-†d(𝐖_t^(N))^† +θ(𝐌_t^(θ))^-†dt.

One can then check directly that (<ref>) solves the SDE (<ref>) by applying Itô's formula to the expression in (<ref>) and using the equations above.





In the setting of Theorem <ref>, we have, for fixed t≥ 0,

    𝐇_t^(N)d=1/2∫_0^t𝐌_s^(ν_N)(𝐌_s^(ν_N))^†ds,

where ν_N=1/4(η+N), with η being the same parameter as throughout this paper.








For any fixed t≥ 0, define 

    (𝐍_s^(ν_N);0≤ s≤ t)def=((𝐌_t^(-ν_N))^-1𝐌_t-s^(-ν_N);0≤ s≤ t).

Then, making the change of variable u→ t-u in (<ref>), one can write

    𝐇_t^(N)d=(𝐌_t^(-ν_N))^-1𝐇_0 (𝐌_t^(-ν_N))^-†+ 1/2∫_0^t𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du.

From (<ref>), we have when 𝐇_0^(N)=0,

    𝐇_t^(N)d=1/2∫_0^t𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du.

Moreover, it is shown in Lemma 2.1 of <cit.> (essentially the matrix analogue of the time-reversal property of Brownian motion) that,

    (𝐍_u^(ν_N);0≤ u≤ t)d=(𝐌_u^(ν_N);0≤ u≤ t),

    which implies the result.




By virtue of (<ref>), it suffices to show that 

    ∫_0^t𝐌_s^(ν_N)(𝐌_s^(ν_N))^†dsd=Π_N^N+1(∫_0^t𝐌_s^(ν_N+1)(𝐌_s^(ν_N+1))^†ds),  ∀ t≥ 0,   N∈ℕ.

By applying Itô's formula we have from (<ref>) with θ=ν_N,

    d(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)   = d𝐌_t^(ν_N)(𝐌_t^(ν_N))^†+𝐌_t^(ν_N)d(𝐌_t^(ν_N))^†+
       d𝐌_t^(ν_N)d(𝐌_t^(ν_N))^†
       =1/2𝐌_t^(ν_N)(d𝐖_t^(N)+d(𝐖_t^(N))^†)(𝐌_t^(ν_N))^†+(N/2-2ν_N)𝐌_t^(ν_N)(𝐌_t^(ν_N))^†dt.

Now, using Lévy's characterization theorem, see <cit.>, one 
can easily check that

    d𝐖̂_t^(N)def=(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)^-1/2𝐌_t^(ν_N)d𝐖_t^(N)(𝐌_t^(ν_N))^†(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)^-1/2,

 is an N× N matrix complex Brownian motion. Thus, the equation (<ref>) can be written in the following closed form:

    d(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)   =1/2√((𝐌_t^(ν_N)(𝐌_t^(ν_N))^†))(d𝐖̂_t^(N)+d(𝐖̂_t^(N))^†)√(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)
         +(N/2-2ν_N)𝐌_t^(ν_N)(𝐌_t^(ν_N))^†dt.

We next consider the SDE for the (N+1) × (N+1) matrix process, that we can write as follows
 
    d𝐌_t^(ν_N+1)   = 1/2𝐌_t^(ν_N+1)d𝐖_t^(N+1) -ν_N+1𝐌_t^(ν_N+1)dt 
       = 1/2[ 𝐌̃_t^(N)      𝐚_t;    𝐛_t^T      c_t ]d[ 𝐖_t^(N)     𝐮_t;   𝐯_t^T     w_t ] 
        - ν_N+1[ 𝐌̃_t^(N)      𝐚_t;    𝐛_t^T      c_t ]dt,

where 𝐌̃_t^(N) is the N× N top-left corner of 𝐌_t^(ν_N+1), 𝐚_t and 𝐛_t are 
ℂ^N-valued, and c_t∈ℂ. Here,
𝐖̃_t^(N), 𝐮_t,𝐯_t, and w_t are complex matrix, complex vector, and ℂ-valued Brownian motions and 𝐀^T denotes the transpose of 𝐀.  One can see from the above equation that

    d[ 𝐌̃_t^(N)      𝐚_t ]=1/2[ 𝐌̃_t^(N)      𝐚_t ]d𝐖_t^(N+1)
        - ν_N+1[ 𝐌̃_t^(N)      𝐚_t ]dt.

We also have that for the N× N top-left corner of 𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†: 

    Π_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†)= [ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†.


Therefore,
by
Itô's formula we get 

    dΠ_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†)   = 
        d[ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†+
      [ 𝐌̃_t^(N)      𝐚_t ]d[ 𝐌̃_t^(N)      𝐚_t ]^†
          + d[ 𝐌̃_t^(N)      𝐚_t ]d[ 𝐌̃_t^(N)      𝐚_t ]
        ^†.

Using (<ref>), we then obtain

    dΠ_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†)     = 1/2[ 𝐌̃_t^(N)      𝐚_t ]( d𝐖_t^(N+1)+ d(𝐖_t^(N+1))
    ^†)
        [ 𝐌̃_t^(N)      𝐚_t ]^†
         + (N+1/2-2ν_N+1)[ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†dt.

Similarly to (<ref>), by virtue of Levy's characterization theorem, we can define the N× N matrix-valued complex Brownian motion (𝐖̂_t^(N))_t≥ 0 (different from (<ref>)) by the formula

    d𝐖̂_t^(N)def=([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)^-1/2[ 𝐌̃_t^(N)      𝐚_t ]d𝐖_t^(N+1)×
    [ 𝐌̃_t^(N)      𝐚_t ]^†([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)^-1/2.

Using these we represent the SDE (<ref>) in the following closed form, recalling (<ref>),

    dΠ_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†) = 1/2√([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)(d𝐖̂_t^(N)+ d(𝐖̂_t^(N))
    ^†)×
       √([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)
        + (N+1/2-2ν_N+1)[ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†dt.

Now, comparing the SDEs (<ref>) and 
(<ref>)
and considering the definition of the parameter ν_N, it follows that, both 
(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†;t ≥ 0)
and
(Π_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†);t ≥ 0)
satisfy the same equation, having a unique weak solution, and hence, these processes have the same distribution. This in particular implies 
that 

    ∫_0^t𝐌_s^(ν_N)(𝐌_s^(ν_N))^†dsd=Π_N^N+1(∫_0^t𝐌_s^(ν_N+1)(𝐌_s^(ν_N+1))^†ds),   N∈ℕ,

 for any fixed t>0, as desired.

We now give a direct proof for Theorem <ref> for any initial condition. 



      We will show  
        
    𝔓_N+1(t)Λ_N^N+1(𝐱,·)=Λ_N^N+1𝔓_N(t)(𝐱,·),  ∀𝐱∈𝕎_N+1,+^∘,  ∀ t≥ 0.

        The case of general 𝐱∈𝕎_N+1,+ then follows by the Feller property, see Proposition <ref>. We have using the explicit expressions in (<ref>) and (<ref>), that (<ref>) boils down to showing:

    ∫_𝕎_N+1,+e^-λ_N+1t(𝔭^N+1_t(x_i,y_j))_i,j=1^N+11_𝐳≺𝐲d𝐲=
            ∫_𝕎_N,+e^-λ_Nt1_𝐲≺𝐱(𝔭^N_t(y_i,z_j))_i,j=1^Nd𝐲.

We write 𝐱≺̃𝐲 to denote y_1≥ x_1> y_2≥ x_2>…≥ x_N> y_N+1, when 𝐱∈𝕎_N and 𝐲∈𝕎_N+1.
Then, we recall the well-known identity
1_𝐱≺̃𝐲= (φ(x_i,y_j))_i,j=1^N+1 with 


    φ(x,y)=
    -1_y>x,       x,y∈ℝ_+,
    
     1,       x=virt,

where we use the convention x_N+1=virt for a “virtual variable". Note that, we can replace 1_𝐲≺𝐱 with 1_𝐲≺̃𝐱 in (<ref>).
Accordingly, with this replacement, the LHS of (<ref>), by the Andreief identity, is equal to 

    e^-λ_N+1t[      [𝔭_t^(N+1)φ^*](x_1,z_1)      [𝔭_t^(N+1)φ^*](x_1,z_2)                            ⋯                            1;      [𝔭_t^(N+1)φ^*](x_2,z_1)      [𝔭_t^(N+1)φ^*](x_2,z_2)                            ⋯                            1;                            ⋮                            ⋮                                                         ⋮; [𝔭_t^(N+1)(t)φ^*](x_N+1,z_1)    [𝔭_t^(N+1)φ^*](x_N+1,z_2)                            ⋯                            1 ],

where φ^*(x,y)=φ(y,x), and by definition [𝔭_t^(N+1)φ^*](x,y) denotes compositition of kernels so that the entries of the matrix above are as follows:

    [𝔭_t^(N+1)φ^*](x_i,z_j)=∫_0^∞𝔭_t^(N+1)(x_i,y)φ^*(y,z_j)dy=-∫_z_j^∞𝔭_t^(N+1)(x_i,y)dy,  ∀ i,j=1,…,N+1.

For the RHS, we first expand the (N+1)×(N+1) determinant corresponding to 1_𝐱≺̃𝐲. We next use the Andreief identity, and then again recombine terms to obtain 

    ∑_l=0^N+1(-1)^N+1-l e^-λ_Nt   ([φ^*𝔭_t^(N)](x_i,z_j))_i,j=1^N+1=
       
         e^-λ_Nt[   [φ^*𝔭_t^(N)](x_1,z_1)   [φ^*𝔭_t^(N)](x_1,z_2)                       ⋯                       1;   [φ^*𝔭_t^(N)](x_2,z_1)   [φ^*𝔭_t^(N)](x_2,z_2)                       ⋯                       1;                       ⋮                       ⋮                                               ⋮; [φ^*𝔭_t^(N)](x_N+1,z_1) [φ^*𝔭_t^(N)](x_N+1,z_2)                       ⋯                       1 ],

where 

    [φ^*𝔭_t^(N)](x_i,z_j)=∫_0^∞φ^*(x_i,y)𝔭_t^(N)(y,z_j)dy=-∫_0^x_i𝔭_t^(N)(y,z_j)dy,  ∀ i,j=1,…,N+1.

We now claim that

    𝔭_t^(N)(x,z)=e^(η/2+N)t∂_x∫_z^∞𝔭_t^(N+1)(x,y)dydef=q_t(x,y).

 We first note, the key algebraic consistency relation
∂_x𝖫_x^(N+1)=𝖫_x^(N)∂_x+(-η/2-N)∂_x.
Making use of this, it can be checked that q_t(x,y) is the fundamental solution to the Kolmogorov backward equation associated to the one-dimensional diffusion
𝖫_x^(N), which is unique, and hence, (<ref>) holds. 
Thus, we have

    -∫_0^x_i𝔭_t^(N)(y,z_j)dy=e^(η/2+N)t[∫_z_j^∞𝔭_t^(N+1)(0,y)dy-∫_z_j^∞𝔭_t^(N+1)(x_i,y)dy], ∀ i,j=1,…,N+1.

One can now substitute this in the determinant in (<ref>) and use column operations to show the desired identity, by noting that λ_N+1-λ_N=N(-η/2-N).






 §.§ Invariant measure and convergence to equilibrium

The inverse Laguerre ensemble 𝔐_N^η
of parameter η>-1 is the probability measure on (𝕎_N,+)_N=1^∞ given by 
	
    𝔐_N^η(d𝐱)=1/𝒵_N^(η)Δ_N^2(x)∏_i=1^Nx_i^-η-2Ne^-1/x_id𝐱,

for some normalisation constant 𝒵_N^(η). We now show that the solution (𝔵^(N)(t);t≥ 0) of (<ref>) started from arbitrary initial condition, converges to 𝔐_N^η.


    Let η>-1 and consider the unique strong solution (𝔵^(N)(t);t≥ 0) of (<ref>). Suppose 𝔎∈ℳ_p(𝕎_N,+)  and that 𝖫𝖺𝗐(𝔵^(N)(0))=𝔎. Then,  as t→∞,

    𝔵^(N)(t) d⟶𝐙,   where 𝖫𝖺𝗐(𝐙)=𝔐_N^η.
 
In particular, 𝔐_N^η is the unique invariant measure of (𝔓_N(t))_t≥ 0.


We use the corresponding matrix process to prove the result.
Recall that from (<ref>), the unique strong solution (𝐇_t^(N);t≥ 0) 
starting from 𝐇_0^(N)∈ℍ_+(N) can be written as 

    (𝐌_t^(ν_N))^-1𝐇^(N)_0 (𝐌_t^(-ν_N))^-†+ 1/2∫_0^t𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du.

Let us take 𝐇_0^(N) such that 𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐇_0^(N)))=𝔎 (and independent of the driving Brownian motions). Then, from Lemma 2.2 in <cit.> the first term converges to 0 almost surely as t→∞. Hence, we obtain

    𝐇_t^(N)d⟶1/2∫_0^∞𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du,  as  t→∞.

Now, by the matrix Dufresne identity proven in <cit.>, see also <cit.>, we know that

    𝖫𝖺𝗐(1/2∫_0^∞𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du) =ℑ𝔏_N^η,

where ℑ𝔏_N^η is the inverse Laguerre measure on ℍ_+(N)  given by the formula,

    ℑ𝔏_N^η(d𝐀)=(𝐀)^-η-2Nexp(-Tr(𝐀^-1))  d𝐀,

with d𝐀 denoting the Lebesgue measure on ℍ_+(N), for some normalisation constant 𝒵̃_N^(η).
By Weyl's integration formula we know that 

    (𝖾𝗏𝖺𝗅_N)_*ℑ𝔏_N^η=𝔐_N^η,

and this concludes the proof of (<ref>). The final statement follows by a standard argument.


As a consequence of Theorem <ref> and uniqueness of the invariant measures from Proposition <ref> we obtain the consistency relation between the inverse Laguerre ensembles; see also <cit.> for a direct proof, using orthogonal polynomials.

     
        Let η>-1. Then, we have
         
    𝔐^η_N+1Λ_N^N+1=𝔐^η_N,  ∀ N∈ℕ.



We also have the following equivalent abstract description of the measure 𝔐^η from the introduction.


Let η>-1. The unique 𝔪∈ℳ_p(Ω_+) satisfying:

    𝔪Λ_N^∞=𝔐_N^η,   ∀ N∈ℕ,

is given by 𝔐^η.



This is the main result of <cit.>.





 §.§ Proof of Theorem <ref> and Theorem <ref>



This follows by taking the semigroups (𝖯_N(t))_t≥ 0 to be (𝔓_N(t))_t≥ 0 in Theorem <ref>, Theorem <ref> and Proposition <ref> all of whose conditions are satisfied by virtue of Proposition <ref> and Proposition <ref> and Theorem <ref> and Theorem <ref>.



By taking (𝖯_N(t))_t≥ 0 to be (𝔓_N(t))_t≥ 0 this is a consequence of Theorem <ref> by virtue of Proposition <ref> and Proposition <ref>. 





 §.§ On the Gibbs property


We briefly discuss a new Gibbs resampling property, see <cit.>, underlying our model. We assume the reader is familiar with the basic terminology of Gibbsian line ensembles, see <cit.>.  This subsection is not part of our argument and can be skipped. 

For η>-1, there exists an explicit spectral expansion of 𝔭_t^(N)(x,y) in terms of special functions, see <cit.>. This expansion is well-adapted to looking at the t→∞ asymptotics of 𝔭_t^(N)(x,y) and using it one can prove that the semigroup (𝔓_N(t))_t≥ 0 is exactly the transition semigroup of N independent one-dimensional diffusions with generator 𝖫^(N) from (<ref>) conditioned to never intersect[We note that this statement does not follow directly from the Doob h-transform structure of (𝔓_N(t))_t≥ 0 in (<ref>). One needs to compute the asymptotics of the first collision time of the N independent one-dimensional diffusions 𝖫^(N) in order show that these two structures coincide and this is in general a delicate problem.]. In particular, (𝔵^(N)(t);t≥ 0), viewed as a line ensemble with N lines, see <cit.>, enjoys a certain Gibbs resampling property with respect to the law of conditioned 𝖫^(N)-bridges, see <cit.> for the rigorous definition of diffusion bridges; in the framework of <cit.> we replace Brownian bridges with 𝖫^(N)-bridges. Similarly, the rescaled process (𝗑^(N)(t);t≥ 0) from (<ref>) enjoys a Gibbs resampling property with respect to 𝖫_rsc^(N)-bridges, where 𝖫_rsc^(N) is the generator of the rescaled by N^-1 diffusion 𝖫^(N) given by:

    𝖫_rsc^(N)=x^2/2∂_x^2+((1-η/2-N)x+1/2N)∂_x,

with transition density denoted by 𝔮_t^(N)(x,y). It is then a natural question whether, the limiting paths ((𝗑_i(t))_i=1^∞;t≥ 0), which we do not yet know if they are non-intersecting, enjoy a Gibbs resampling property as well. If they did this would give an alternative, to the one in Section <ref> below, albeit more complicated, way to prove non-intersection.

One can show, that given the convergence of the paths, that we already know from Theorem <ref>, this question really boils down, modulo assumptions of lesser importance to avoid pathological situations, to convergence of the free 𝖫_rsc^(N)-bridge, whose time-inhomogeneous generator, in case of the bridge ending at point y at time T>0, is given by, for 0≤ t<T, see <cit.>,

    x^2/2∂_x^2+((1-η/2-N)x+1/2N+x^2∂_x log𝔮_T-t^(N)(x,y))∂_x,

to a limiting diffusion bridge as N →∞. This question turns out to be complicated. The spectral expansion in <cit.> is not well-adapted for looking at the N →∞ limit of the above expression.  Nevertheless, using some judicious transformations, it is possible to connect this problem to a problem of asymptotics of the logarithmic derivative of the transition kernel of a Schrödinger semigroup with an explicit N-dependent Morse potential 𝖵_N(x), see <cit.>. What we gain by doing this is that all N-dependence of the problem is now in the potential 𝖵_N(x) and it is  easy to see that it converges to an explicit potential 𝖵_∞(x). Alas, the limiting potential 𝖵_∞(x) has the “wrong" sign and the associated Schrödinger semigroup blows up! In particular, the transition kernel of the Schrödinger semigroup with potential 𝖵_N(x) diverges as N→∞. However, its logarithmic derivative may still well converge, due a subtle cancellation, and so this does not contradict the  existence of a limiting Gibbs property. 









§ THE ISDE


In this section we first prove that (𝗑_i(·))_i=1^∞ almost surely consists of non-intersecting paths in Theorem <ref> and then in Theorem <ref> that it solves the ISDE (<ref>). The main idea will be to use various “characteristic polynomial"-like objects associated to the finite-dimensional dynamics (𝗑^(N)_i(·))_i=1^N, first as Lyapunov functions, to control uniformly in N, the times two paths come close and then to show convergence of the singular drift term. We need some preliminaries.

We recall a result from <cit.> on convergence of polynomials with real zeros to analytic functions that we use as one of the ingredients in the proofs of 
Theorem <ref> and Theorem <ref> below. Given, 𝐱^(N)∈𝕎_N define the polynomial

    𝔘_N(z;𝐱^(N))def=∏_j=1^N(1-x_j^(N)z).

This is the reverse characteristic polynomial of a matrix with eigenvalues 𝐱^(N). We also have the following definition.






Define the entire functions 𝖤(z;ω) and 𝖤_+(z;ω), indexed by ω∈Ω and ω∈Ω_+ respectively,
given by their Hadamard products:

    𝖤(z;ω)   def=e^-γ z-δ-∑_i=1^∞[(x_i^+)^2+(x_i^-)^2]/2z^2∏_j=1^∞e^x_j^+ z(1-x_j^+z)∏_j=1^∞e^-x_j^- z(1+x_j^- z),  ω∈Ω,
    𝖤_+(z;ω)   def=e^-γ z∏_j=1^∞e^x_j z(1-x_jz),  ω∈Ω_+.

  


Then, we have the following result, see
Proposition 2.5 and Remark 2.7 of <cit.>.


Suppose (𝐱^(N))_N=1^∞∈(𝕎_N)_N=1^∞ such that ω(𝐱^(N))N→∞⟶ω∈Ω. Then, the following convergence holds uniformly on compact sets in ℂ:

    𝔘_N(z;𝐱^(N))→𝖤(z;ω),    as  N→∞.

If instead (𝐱^(N))_N=1^∞∈(𝕎_N,+)_N=1^∞ such that ω(𝐱^(N))N→∞⟶ω∈Ω_+, then the following convergence holds uniformly on compact sets in ℂ:

    𝔘_N(z;𝐱^(N))→𝖤_+(z;ω),    as  N→∞.


Note that this convergence statement is consistent with the embedding (<ref>) of Ω_+ into Ω. Below we will only use the result for 𝕎_N,+ and Ω_+.






 §.§ Non-intersection via characteristic polynomials




We use the following notation convention throughout. For (𝐱,γ)∈Ω_+ we write ℙ_(𝐱,γ) for the law of (𝐗_t^Ω_+;t≥ 0) from Theorem <ref> starting from (𝐱,γ) and abusing notation (as we do not assume that all these processes are defined on the same probability space unless otherwise stated), for 𝐱^(N)∈𝕎_N,+, we write ℙ_𝐱^(N) for the law of the process solving (<ref>) starting from 𝐱^(N). We write 𝔼_(𝐱,γ) and 𝔼_𝐱^(N) for the corresponding expectations.


   Let η∈ℝ, 𝐱∈𝕎_∞,+^∘, γ∈ℝ_+, with ∑_i=1^∞ x_i≤γ. Consider the Feller process on Ω_+, (𝐗_t^Ω_+;t≥ 0)=(((𝗑_i(t))_i=1^∞,γ(t));t≥ 0) from Theorem <ref> with initial condition 𝐗_0^Ω_+=(𝐱,γ). Then, for all T≥ 0 and n∈ℕ we have
    
    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_n(t)>0
    )   =1,
    ℙ_(𝐱,γ)(inf_t∈[0,T]|
    𝗑_n(t)-𝗑_n+1(t)
    |>0)   =1.




  We prove the result using an inductive argument. For each N∈ℕ, let 𝐱^(N)∈𝕎_N,+^∘ converge to (𝐱,γ)∈Ω_+ under the embedding (<ref>). 
Let 𝗑^(N) be the solution to the equation (<ref>) started at 𝐱^(N).
We will make use of the log transform of (<ref>), which we consider in its integral form:

    log𝗑_i^(N)(t)    = log x_i^(N)+ 𝗐_i(t) -1+η/2t+∫_0^t1/2N𝗑_i^(N)(s)ds
         +
    ∫_0^t∑_j=1, j≠ i^N𝗑_j^(N)(s)/𝗑_i^(N)(s)-𝗑_j^(N)(s)ds, i=1,…,N.

Let us continue with some definitions. We always use the convention inf∅ =∞ in what follows. 
Define, 
for any N∈ℕ, the following stopping times (with respect to the natural filtration) associated to (𝗑_i^(N))_i=1^N,

    σ_n^N(ϵ)   def=inf{t≥ 0;𝗑_n^(N)(t)≤ϵ},  ϵ>0,  n=1,…,N
    τ_n^N(δ)   def=inf{t≥ 0;|1-𝗑_n+1^(N) (t)/𝗑_n^(N) (t)|≤δ},  δ>0,  n=1,…,N-1,
    θ_n^N(r)   def=inf{t≥ 0;𝗑_n^(N)(t)≥ r},  r>0,  n=1,…,N,

and 
analogously, the stopping times (with respect to the natural filtration) associated to (𝗑_i)_i=1^∞,

    σ_n(ϵ)   def=inf{t≥ 0;𝗑_n(t)≤ϵ},  ϵ>0,  n∈ℕ,
    τ_n(δ)   def=inf{t≥ 0;|1-𝗑_n+1 (t)/𝗑_n (t)|≤δ},  δ>0,  n∈ℕ,
    θ_n(r)   def=inf{t≥ 0;𝗑_n(t)≥ r},  r>0,  n∈ℕ.

Note that, by Lemma <ref>, τ_n^N(δ) is well-defined for any N∈ℕ and n≤ N-1. 
It will be justified later on that τ_n(δ) is also well-defined for all n∈ℕ.


Now, let N∈ℕ. We define for any 𝐱^(N)∈𝕎_N,+^∘, the following variant of the characteristic polynomial associated to 𝐱^(N),

    Ψ_n^N(𝐱^(N))   def=∏_i=1^n∏_j=n+1^N(1-x_j^(N)/x_i^(N)),   n=1,…,N-1,
 
and consider the Lyapunov function
given as:

    f_n^N(𝐱^(N))def=-log(Ψ_n^N(𝐱^(N))),   n=1,…,N-1.

It is easy to see that f_n^N(𝐱^(N))>0 for all 𝐱^(N)∈𝕎_N,+^∘ and n=1,…,N-1.
    We next define
 
    κ_n^N(R)def=inf{t≥ 0;f_n^N(𝗑^(N)(t))≥ R},    R>0,   n=1,…,N-1.

    Observe that,

    {τ_n^N(δ)≤ a}⊂{κ_n^N(log(δ^-1))≤ a},  for δ<1.

We are going to prove by induction on n that 
for all T≥ 0 and n∈ℕ, 

    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_n(t)>0
    )   =1,
    ℙ_(𝐱,γ)(inf_t∈[0,T]|1-𝗑_n+1(t)/𝗑_n(t)|>0)   =1.

Note that, this implies non-collision, since given (<ref>) and (<ref>), we have almost surely,

    inf_t∈[0,T]|
    𝗑_n(t)-𝗑_n+1(t)
    |≥inf_t∈[0,T]𝗑_n(t)
    inf_t∈[0,T]|1-𝗑_n+1(t)/𝗑_n(t)|,

and hence,

    ℙ_(𝐱,γ)(inf_t∈[0,T]|
    𝗑_n(t)-𝗑_n+1(t)
    |>0)=1.


The base case.
Using (<ref>), we have for any t≥ 0 that
	
    log𝗑_1^(N)(t∧σ_1^N(ϵ)∧θ_1^N(r))   ≥log x_1^(N)
    		-|1+η/2|t+𝗐_1(t∧σ_1^N(ϵ)∧θ_1^N(r)).

	Therefore, since t ↦𝗐_1(t∧σ_1^N(ϵ)∧θ_1^N(r)) is a mean-zero martingale, taking expectation of both sides we obtain
	
    log r+logϵℙ_𝗑^(N)(σ_1^N(ϵ)< t∧θ_1^N(r))   ≥log x_1^(N)-|1+η/2|t,

and thus, for any ϵ<1,

    ℙ_𝐱^(N)(σ_1^N(ϵ)< t∧θ_1^N(r))   ≤log r-log x_1^(N)+|1+η/2|t/|logϵ|.

Observe that, the bound (<ref>) is uniform in N (recall that x_1^(N)→ x_1>0). We would like to conclude that the same bound (with x_1 in the formula instead) holds for the stopping times σ_1(ϵ) and θ_1(r) corresponding to the limit paths. A little argument is needed though. These stopping times, viewed as functionals on paths are not continuous, but fortunately they are not far from continuous. For more details see Chapter VI, Sections 2,3 of <cit.> which we follow closely. For a ∈ℝ_+, f∈ C(ℝ_+,ℝ_+), we define

    𝖲_≥^a(f)=inf{t≥ 0; f(t)≥ a},  𝖲_≤^a(f)=inf{t≥ 0; f(t)≤ a}.

 
 Write ℝ_+,ext=ℝ_+∪{∞}. For fixed f∈ C(ℝ_+,ℝ_+) define the map 𝖡_f by,
 
    𝖡_f:ℝ_+^2    ⟶ℝ_+,ext^2
    
        (a,b)   ↦(𝖲_≤^a(f),𝖲_≥^b(f)),

 which from Lemma 2.10 of Chapter VI of <cit.> only has jump discontinuities, while for fixed (a,b)∈ℝ_+^2 define the map 𝖠_(a,b) by,
 
    𝖠_(a,b):C(ℝ_+,ℝ_+)    ⟶ℝ_+,ext^2
    
     f   ↦(𝖲_≤^a(f),𝖲_≥^b(f)).

 Now, if we let 𝔇_(a,b) denote the set of discontinuities of the map 𝖠_(a,b) and 𝖣(f) denote the set of discontinuities of the map 𝖡_f, then from Proposition 2.11 of Chapter VI of <cit.>, since 𝖠_(a,b) is continuous at every f so that (a,b)∉𝖣(f), we obtain that,
 
    𝔇_(a,b)⊂{f∈ C(ℝ_+,ℝ_+):(a,b)∈𝖣(f)}.

 Let us write ℙ^(1)_(𝐱,γ) for 𝖫𝖺𝗐(𝗑_1) on C(ℝ_+,ℝ_+) under ℙ_(𝐱,γ). Then, define

    𝒱={(a,b)∈ℝ_+^2:ℙ_(𝐱,γ)^(1)(𝔇_(a,b))>0}⊂{(a,b)∈ℝ_+^2:ℙ_(𝐱,γ)^(1)((a,b)∈𝖣(𝗑_1))>0}.
 
Then, by virtue of the fact that the map 𝖡_f only has jump discontinuities, we obtain, see Lemma 3.12 of Chapter VI of <cit.>, that the sets 𝒱_1={a∈ℝ_+:(a,b)∈𝒱} and 𝒱_2={b∈ℝ_+:(a,b)∈𝒱} are at most countable. In particular, we can find sequences (ϵ_m)_m=1^∞ and (r_ℓ)_ℓ=1^∞, with ϵ_m→ 0 and r_ℓ→∞, as m,ℓ→∞, so that for any m,ℓ∈ℕ, ℙ_(𝐱,γ)^(1)(𝔇_(ϵ_m,r_ℓ))=0. Now, observe that,

    (σ_1^N(ϵ),θ_1^N(r))=𝖠_(ϵ,r)(𝗑_1^(N))
    ,   (σ_1(ϵ),θ_1(r))=𝖠_(ϵ,r)(𝗑_1).

Hence, by the continuous mapping theorem, we obtain that for all m,l∈ℕ,  as N →∞,

    (σ_1^N(ϵ_m),θ_1^N(r_ℓ))d⟶(σ_1(ϵ_m),θ_1(r_ℓ)).

By the Portmanteau theorem for convergence in distribution, we get that, for any open set 𝒪⊆ℝ^2,  
 
    ℙ_(𝐱,γ)((σ_1(ϵ_m),θ_1(r_ℓ))∈𝒪)
         ≤lim inf_N→∞ℙ_𝐱^(N)((σ_1^N(ϵ_m),θ_1^N(r_ℓ))∈𝒪).

 From this and  (<ref>) we conclude that (for ϵ_m<1),

    ℙ_(𝐱,γ)(σ_1(ϵ_m)< t∧θ_1(r_ℓ))≤lim inf_N→∞ℙ_𝗑^(N)(σ_1^N(ϵ_m)< t∧θ_1^N(r_ℓ))≤log r_ℓ-log x_1+|1+η/2|t/|logϵ_m|,

and hence, by the monotone convergence theorem,

    ℙ_(𝐱,γ)(lim_m→∞σ_1(ϵ_m)< t∧θ_1(r_ℓ))=0.

Therefore, by taking ℓ,t→∞, it follows, by virtue of the fact that (recall that r_ℓ→∞), θ_1(r_ℓ)a.s.⟶∞, that ℙ_(𝐱,γ)(lim_m →∞σ_1(ϵ_m)=∞)=1, namely, since ϵ_m → 0,

    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_1(t)>0
    )=1.



We now proceed to prove the second assertion. 
Using Itô's formula, we have 

    df_1^N(𝗑^(N)(t))   =
        d𝖬_1^(N)(t)+
        N-1/2N𝗑_1^(N)(t)dt-∑_j=2^N(𝗑_j^(N)(t))^2/(𝗑_1^(N)(t)-𝗑_j^(N)(t))^2dt 
         
    +∑_j=2^N∑_ℓ=2,ℓ≠ j^N𝗑_j^(N)(t)𝗑_ℓ^(N)(t)/(𝗑_1^(N)(t)-𝗑_ℓ^(N)(t))(𝗑_j^(N)(t)-𝗑_ℓ^(N)(t))dt.

where

    𝖬_1^(N)(t)def=(N-1) 𝗐_1(t)+∑_ℓ=2^N∫_0^t𝗑_1^(N)d𝗐_1(s)-𝗑_ℓ^(N)d𝗐_ℓ(s)/𝗑_1^(N)-𝗑_ℓ^(N),   ∀ t ≥ 0,

is a local martingale. Now, we observe, and this is key, that for any t≥ 0, the last term on the RHS of (<ref>) is non-positive, by virtue of the elementary inequality, for 𝐲∈𝕎_N,+^∘,

    ∑_j=2^N∑_ℓ=2,ℓ≠ j^Ny_j y_ℓ/(y_1-y_ℓ)(y_j-y_ℓ)
    =∑_j<ℓj,ℓ=2^Ny_j y_ℓ/y_j-y_ℓ(1/y_1-y_ℓ-1/y_1-y_j)≤ 0.

Thus, we can drop this term and write 

    f_1^N(𝗑^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)))   ≤ f_1^N(𝐱^(N))+𝖬_1^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)) 
         +∫_0^t∧κ_1^N(R)∧σ_1^N(ϵ)1/2𝗑_1^(N)(s)ds
       ≤ f_1^N(𝐱^(N))+𝖬_1^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ))+ 
        t/ϵ.

Now, since the stopped process t ↦𝖬_1^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)) is a mean-zero martingale, we get

    𝔼_𝐱^(N)[f_1^N(𝗑^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)))]   ≤ f_1^N(𝐱^(N))+t/ϵ.

Therefore, we have

    Rℙ_𝐱^(N)(κ_1^N(R)< t∧σ_1^N(ϵ))≤sup_N∈ℕf_1^N(𝐱^(N))+
        t/ϵ.

Note that, since
ω(𝐱^(N))N→∞⟶(𝐱,γ), it follows from Proposition <ref> that 
Ψ_1^N(𝐱^(N)) converges to a strictly positive limit, 
and hence, 
Cdef=sup_N∈ℕf_1^N(𝐱^(N))<∞. We thus have,

    ℙ_𝐱^(N)(κ_1^N(R)< t∧σ_1^N(ϵ))≤C+t/ϵ/R.

Let δ<1 be arbitrary. We have in view of (<ref>) that 

    ℙ_𝐱^(N)(τ_1^N(δ)< t∧σ_1^N(ϵ))≤C+t/ϵ/|logδ|,

and note that this bound is independent of N. Observe that,

    τ_1^N(δ)=𝖲_≤^δ(|1-𝗑_2^(N)/𝗑_1^(N)|),   τ_1(δ)=𝖲_≤^δ(|1-𝗑_2/𝗑_1|),

and that by virtue of (<ref>), t↦ |𝗑_1(t)|^-1|𝗑_1(t)-𝗑_2(t)| is almost surely, a well-defined continuous function. We now argue similarly to before to obtain the existence of sequences, abusing notation (the sequence (ϵ_ℓ)_ℓ=1^∞ here may have nothing to do with the sequence (ϵ_m)_m=1^∞ from earlier), (δ_m)_m=1^∞, (ϵ_ℓ)_ℓ=1^∞, with δ_m, ϵ_ℓ→ 0, as m,ℓ→∞, so that, for all m,ℓ∈ℕ, as N →∞,

    (τ_1^N(δ_m),σ_1^N(ϵ_ℓ))d⟶(τ_1(δ_m),σ_1(ϵ_ℓ)).

Hence, by taking N→∞ in (<ref>), one obtains, for δ_m<1,

    ℙ_(𝐱,γ)(τ_1(δ_m)< t∧σ_1(ϵ_ℓ))≤lim inf_N →∞ℙ_𝐱^(N)(τ_1^N(δ_m)< t∧σ_1^N(ϵ_ℓ))≤C+t/ϵ_ℓ/|logδ_m|.

It thus follows that,

    ℙ_(𝐱,γ)(lim_m→∞τ_1(δ_m)< t∧σ_1(ϵ_ℓ))=0.

Now, we take t→∞,
 and then ℓ→∞, to finally get

    ℙ_(𝐱,γ)(lim_m→∞τ_1(δ_m)≥lim_ℓ→∞σ_1(ϵ_ℓ))=1.

But ℙ_(𝐱,γ)(lim_ℓ→∞σ_1(ϵ_ℓ )=∞)=1 by virtue of (<ref>).
This implies

    ℙ_(𝐱,γ)(lim_m →∞τ_1(δ_m)=∞)=1.

That is, for all T≥ 0, we have

    ℙ_(𝐱,γ)(inf_t∈[0,T]|1-𝗑_2(t)/𝗑_1(t)|>0)=1,

which completes the proof of the case n=1.

The inductive step. We next assume that (<ref>) and  (<ref>) hold for n=k, and prove them for n=k+1.
From (<ref>), we can write

    log𝗑_k+1^(N)(t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r))    ≥log x_k+1^(N)
        +𝗐_k+1(t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r))
     
     -|1+η/2|   t+∫_0^t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r)∑_j=1, j≠ k+1^k𝗑_j^(N)(t)/𝗑_k+1^(N)(t)-𝗑_j^(N)(t)dt
    ≥log x_k+1^(N)   
        -|1+η/2|t+𝗐_k+1(t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r))-kt/δ,

 and thus, we have,

    log r+logϵℙ_𝗑^(N)(σ_k+1^N(ϵ)< t∧τ_k^N(δ)∧θ_k+1^N(r))   ≥log x_k+1^(N)-|1+η/2|t-kt/δ.

 One can check, by making use of the induction hypothesis, that as previously, there exist sequences (ϵ_m)_m=1^∞, (δ_ℓ)_ℓ=1^∞, (r_q)_q=1^∞, with ϵ_m,δ_ℓ→ 0, r_q →∞, as m,ℓ,q →∞, so that the corresponding stopping times converge in distribution, as N →∞, to their limiting counterparts, and in particular we obtain,
 
    ℙ_(𝐱,γ)(lim_m→∞σ_k+1(ϵ_m)< t∧τ_k(δ_ℓ)∧θ_k+1(r_q))=0.

 Hence, it follows that ℙ_(𝐱,γ)(lim_m →∞σ_k+1(ϵ_m)=∞)=1, if we take m →∞ first, and then, t,ℓ,q→∞ whereby τ_k(δ_ℓ),θ_k+1(r_q)a.s.⟶∞. Equivalently,

    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_k+1(t)>0
    )=1.

 
We next apply Itô's formula to f_k+1^N to obtain

    df_k+1^N(𝗑^(N)(t))   =
        d𝖬_k+1^(N)(t)+
        ∑_i=1^k+1N-k-1/2N𝗑_i^(N)(t)dt-∑_i=1^k+1∑_j=k+2^N(𝗑_j^(N)(t))^2/(𝗑_i^(N)(t)-𝗑_j^(N)(t))^2dt  
         +
       ∑_i=1^k+1∑_j=k+2^N∑_ℓ≠ i,jℓ=1^N
    𝗑_j^(N)(t)𝗑_ℓ^(N)(t)/(𝗑_i^(N)(t)-𝗑_ℓ^(N)(t))(𝗑_j^(N)(t)-𝗑_ℓ^(N)(t))dt,

where t↦𝖬_k+1^(N)(t) stands for the corresponding local martingale term.
We note, and this is again important, that the last term in the above expression is non-positive for any t≥ 0, by virtue of the inequality, for 𝐲∈𝕎_N,+^∘,

    ∑_i=1^k+1∑_j=k+2^N∑_ℓ≠ i,jℓ=1^N y_j y_ℓ/(y_i-y_ℓ)(y_j-y_ℓ)≤ 0,

which is easily seen to be true since the LHS can be written as,

    ∑_i=1^k+1∑_j=k+2^N∑_ℓ=1,ℓ≠ i^k+1y_j y_ℓ/(y_i-y_ℓ)(y_j-y_ℓ) +∑_i=1^k+1∑_j=k+2^N∑_ℓ≠ jℓ=k+2^N y_j y_ℓ/(y_i-y_ℓ)(y_j-y_ℓ)
       =
    ∑_j=k+2^N∑_i<ℓi,ℓ=1^k+1y_j/y_i-y_ℓ(y_ℓ/y_j-y_ℓ-y_i/y_j-y_i) +∑_i=1^k+1∑_j<ℓj,ℓ=k+2^Ny_j y_ℓ/y_j-y_ℓ(1/y_i-y_ℓ-1/y_i-y_j) ≤ 0.

Thus, by dropping it, we have

    f_k+1^N(𝗑^(N)(t∧κ_k+1^N(R)∧σ_k+1^N(ϵ)))   ≤ f_k+1^N(𝐱^(N))+𝖬_k+1^(N)(t∧κ_k+1^N(R)∧σ_k+1^N(ϵ))
         +∑_i=1^k+1∫_0^t∧κ_k+1^N(R)∧σ_k+1^N(ϵ)1/2𝗑_n^(N)(t)dt
       ≤  f_k+1^N(𝐱^(N))+𝖬_k+1^(N)(t∧κ_k+1^N(R)∧σ_k+1^N(ϵ))+ 
        k/ϵt.

One can now follow the argument for the case n=1 to establish the desired result in the exact same way.



We note that in the argument above we only needed to take as input convergence of the paths and not necessarily convergence in C(ℝ_+,Ω_+). Had we not even known convergence of the paths, the argument would in any case give that any possible subsequential limits would need to consist of non-intersecting paths.





 §.§ The ISDE via characteristic polynomials


We now prove that (𝗑_i(·))_i=1^∞ solves the ISDE (<ref>). Beyond the use of characteristic polynomials to show convergence of the singular drift, it is interesting to note the following. In the construction of weak solutions to finite-dimensional SDE, see <cit.>, the driving Brownian motions can usually[The upshot is that one wants to construct the driving Brownian motions from the solution, see <cit.>. If the noise degenerates, one needs an auxiliary, independent from the solution, Brownian motion, see <cit.>. However, in our setting the noise does not degenerate as 𝗑_i>0 and this is not the source of the additional randomness. ] be taken to be measurable with respect to the natural filtration generated by the solution. Here, the driving Brownian motions (𝗐_i(·))_i=1^∞ will in fact be measurable not with the filtration generated by (𝗑_i(·))_i=1^∞ on its own but rather with the filtration generated by the enhanced process 𝐗_·^Ω_+=((𝗑_i(·))_i=1^∞,γ(·)). This is another illustration of the phenomenon that in the infinite-dimensional limit additional information is created.


    Let η∈ℝ, 𝐱∈𝕎_∞,+^∘, γ∈ℝ_+ with ∑_i=1^∞x_i≤γ.
    Let 𝐗_·^Ω_+ and 𝗑(·)=(𝗑_i(·))_i=1^∞ be as in Theorem <ref>. Then, 𝗑(·) is a weak solution, starting from 𝐱, to the ISDE (<ref>). Moreover, we have that,
 
    ℙ_(𝐱,γ)(𝐗^Ω_+_t∈Ω_+^0, ∀ t>0)=1,   ∀ (𝐱,γ)∈Ω_+.






We first prove that there exists a filtered probability space, on which independent standard Brownian motions (𝗐̃_i)_i=1^∞ adapted to the filtration and an equal in law copy (𝗑̃,γ̃) of (𝗑,γ), also adapted to the filtration, are defined, such that almost surely, for all t≥ 0:
   	
    𝗑̃_i(t) = x_i+ ∫_0^t𝗑̃_i(s) d𝗐̃_i(s) +∫_0^t (-η/2𝗑̃_i(s)+γ̃(s)-∑_j=1^∞𝗑̃_j(s)+∑_j=1,j≠ i^∞𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s))ds,    i∈ℕ.

The filtration will be the natural filtration of (𝗑̃,γ̃). Recall that the natural filtration (ℱ_t)_t≥ 0 of a process (ℜ_t;t≥ 0) is given by, for all t≥ 0,

    ℱ_t=σ(ℜ_t; 0 ≤ s ≤ t),

 where for a collection of random variables 𝒞, we write σ (𝒞) for the sigma algebra generated by them. 
 
We now prove this claim. For each N∈ℕ, let 𝐱^(N)∈𝕎_N,+^∘ be such that it converges, under the embedding (<ref>), to (𝐱,γ).  Let 𝗑^(N) be the solution to the equation (<ref>) starting at 𝐱^(N). 
Writing the equation in the integral form, one has for i=1,2,…,N,

    𝗑_i^(N)(t)  =	x_i^(N) +∫_0^t𝗑_i^(N)(s) d𝗐_i(s) - η/2∫_0^t𝗑_i^(N)(s)ds
    	+t/2N
    	+∫_0^t∑_j=1,j≠ i^N𝗑_i^(N)(s)𝗑_j^(N)(s)/𝗑_i^(N)(s)-𝗑_j^(N)(s)ds.
   
  By Theorem <ref>, 𝐗^(N)d⟶(𝗑,γ) on C(ℝ_+, Ω_+) where as usual, 𝐗^(N) corresponds to the embedded process on Ω_+. Now, by the Skorokhod
 representation theorem, there exists a probability space (Ω̃,𝐏̃) on which 
 𝗑̃^(N)d=𝗑^(N), i.e.  𝐗̃^(N)d=𝐗^(N), and 𝐗̃^Ω_+d=𝐗^Ω_+, i.e. (𝗑̃,γ̃)d=(𝗑,γ), 
  are defined such that 𝐏̃-a.s.,
  
    𝐗̃^(N)N→∞⟶ (𝗑̃,γ̃),  on   C(ℝ_+, Ω_+).

Let us fix N ∈ℕ. Let us write (ℱ_t^(N))_t≥ 0 for the natural filtration of 
𝗑̃^(N). We first show that there exist  (ℱ_t^(N))_t≥ 0-adapted independent standard Brownian motions (𝗐̃_i^(N))_i=1^N so that 𝐏̃-a.s., for all t≥ 0 and  i=1,2,…,N, we have
 
    𝗑̃_i^(N)(t)  =	x_i^(N)+∫_0^t𝗑̃_i^(N)(s) d𝗐̃_i^(N)(s) - η/2∫_0^t𝗑̃_i^(N)(s)ds
      +t/2N
       +∫_0^t∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)ds.
 
Let us define the process ((𝖸_i^(N)(t))_i=1^N;t ≥ 0) coordinate-wise, as follows,

    (𝖸_i^(N)(t);t≥ 0)   def=(𝗑̃_i^(N)(t) -	x_i^(N) + η/2∫_0^t𝗑̃_i^(N)(s)ds
      -t/2N
       -∫_0^t∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)ds;t≥ 0)
       d=(∫_0^t 𝗑_i^(N) (s)d𝗐_i(s) t≥ 0).

The equality in distribution is by virtue of the fact that 𝗑̃^(N)d=𝗑^(N) and that 𝗑^(N) is a solution to the SDE (<ref>). Observe that, 𝖸^(N) is adapted with respect to (ℱ_t^(N))_t≥ 0. It is also easy to check that 𝖸^(N) is an N-dimensional continuous local martingale with respect to 
(ℱ_t^(N))_t≥ 0, with quadratic variation[For a continuous N-dimensional local martingale (𝗓_i(·))_i=1^N we write ⟨𝗓_i, 𝗓_j ⟩_t, i,j=1,…,N for its quadratic (co-)variation.]  <cit.> given by,

    ⟨𝖸_i^(N),𝖸_j^(N)⟩_t=1_i=j∫_0^t 𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)ds,   ∀ t ≥ 0,  i,j=1,2,…,N.

Now, we can define, for i=1,…,N, by virtue of the fact that 𝐏̃-a.s. 𝗑̃_i^(N)>0, the stochastic integral,

    𝗐̃_i^(N)(t)def=∫_0^t 1/𝗑̃_i^(N)(s)d𝖸_i^(N)(s),   ∀ t ≥ 0.

Observe that, (𝗐̃_i^(N))_i=1^N is adapted with respect to (ℱ_t^(N))_t≥ 0 and moreover it is in fact an N-dimensional continuous local martingale, with respect to this filtration, satisfying,

    ⟨𝗐̃_i^(N),𝗐̃_j^(N)⟩_t=1_i=jt,   ∀ t ≥ 0,  i,j=1,2,…,N.

Hence, by virtue of Levy's characterisation theorem <cit.>, (𝗐̃_i^(N))_i=1^N is a sequence of independent standard Brownian motions. Moreover, from (<ref>), we have

    𝖸_i^(N)(t)=∫_0^t 𝗑̃_i^(N)(s)d𝗐̃_i^(N)(s),   ∀ t ≥ 0,  i=1,2,…,N,

from which (<ref>) follows.








We next prove (<ref>) holds. 
Clearly, proving (<ref>) amounts to showing the convergence of the integrals to the corresponding terms in the limit. Let i∈ℕ be fixed. We proceed to show the convergence of the drift terms first. Observe that, 𝐏̃-a.s., for all t≥ 0,

    ∫_0^t𝗑̃_i^(N)(s)ds→∫_0^t𝗑̃_i(s)ds,    as  N→∞.

We thus need to prove convergence of the interaction terms, namely, 𝐏̃-a.s., for all t ≥ 0,

    ∫_0^t∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)dsN→∞⟶1/2∫_0^t(γ̃(s)-∑_j=1^∞𝗑̃_j(s))ds
       +∫_0^t∑_j=1,j≠ i^∞𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

To this end, we will use Proposition <ref> on convergence of the corresponding characteristic polynomials.
Let N∈ℕ and 1≤ i≤ N. We consider the following entire functions: 
    Φ_i^N(z;𝐱^(N))   def=∏_j=1,j≠ i^N(1-x_j^(N)z)^2,  𝐱^(N)∈𝕎_N,+,
    𝖤_i(z;ω)   def=e^-2(γ-x_i) z∏_j=1,j≠ i^∞e^2x_jz(1-x_jz)^2,  ω=(𝐱,γ)∈Ω_+.

Observe that, the integrands in (<ref>) are determined in terms of the functions Φ_i^N(z;𝐱^(N)) and 𝖤_i(z;ω) respectively. Indeed, 
the corresponding terms, at time t≥ 0, are equal to 

    1/2d/d zlogΦ_i^N(z,𝗑̃^(N)(t))|_z=(𝗑̃_i^(N)(t))^-1,
       1/2d/d zlog𝖤_i(z;𝐗̃_t^Ω_+)|_z=(𝗑̃_i(t))^-1.

Note that, the expressions above are well-defined by virtue of the fact that the paths are 𝐏̃-a.s. non-intersecting and never hit zero. We now prove that, 𝐏̃-a.s., for all t≥ 0,

    d/d zlogΦ_i^N(z,𝗑̃^(N)(t))|_z=(𝗑̃_i^(N)(t))^-1N→∞⟶d/dzlog𝖤_i(z;𝐗̃_t^Ω_+)|_z=(𝗑̃_i(t))^-1.

 Since we have convergence on Ω_+, as in (<ref>), it follows from
 Proposition <ref>, in particular equation (<ref>), that 𝐏̃-a.s., for all t≥ 0,

    Φ_i^N(z,𝗑̃^(N)(t))⟶𝖤_i(z;𝐗̃_t^Ω_+),   as  N→∞,

uniformly on compact sets in ℂ, and the same is true for the corresponding derivatives by analyticity.
Note that, by virtue of Theorem <ref> we have 𝐏̃-a.s., for all t≥ 0

    (𝗑̃_i^(N)(t))^-1⟶(𝗑̃_i(t))^-1,   as N→∞.

From the above, and the fact that all processes we are considering are 𝐏̃-a.s. non-intersecting, see Lemma <ref> and Theorem <ref>, the convergence in (<ref>) is concluded. This shows convergence of the integrands for all times t≥ 0. We now prove that the corresponding integrals also converge. First, note that as the paths are ordered and 𝗑̃_i^(N)> 0, we can write, 𝐏̃-a.s., for all t≥ 0,
     
    𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)≤𝗑̃_i+1^(N)(s)/𝗑̃_i^(N)(s),  ∀ j,N≥ i+1,  s∈[0,t].

    Also, using convergence of the coordinates and since almost surely 𝗑̃_i> 0, we have that 𝐏̃-a.s., for all t≥ 0,
    
    inf_s∈[0,t]|1-𝗑̃_i+1^(N)(s)/𝗑̃_i^(N)(s)|→inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)| ,   as  N→∞.

    On the other hand, by the non-intersection property of the limiting paths, 𝐏̃-a.s., for all t≥ 0,
    
    inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|>0.

    Hence, there exists a random N_0∈ℕ such that, 𝐏̃-a.s., for all t≥ 0,
        
    inf_s∈[0,t]|1-𝗑̃_i+1^(N)(s)/𝗑̃_i^(N)(s)|≥1/2inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|,   ∀ N≥ N_0.

    Observe now that for all N≥ N_0, we have 𝐏̃-a.s., for all t≥ 0,
    
    𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)≤ 2(inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|)^-1𝗑̃_j^(N)(s),  ∀ j≥ i+1,  s∈[0,t],
 
    and therefore, 𝐏̃-a.s., for all t≥ 0
       
    ∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)≤ 2(inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|)^-1∑_j=1^N𝗑̃_j^(N)(s),   ∀ N≥ N_0.

This, together with the fact that 𝐏̃-a.s., ∑_j=1^N𝗑̃_j^(N)(·)N→∞⟶γ̃(·) in C(ℝ_+,ℝ_+) implies (<ref>) by applying the generalized Lebesgue dominated convergence theorem. 

We now deal with the stochastic integral term. From now on, let us denote by (ℱ_t)_t≥ 0 the natural filtration of the process (𝐗̃^Ω_+_t; t≥ 0). We deduce from the convergence of both the LHS and the drift term in (<ref>) that 𝐏̃-a.s., for all i∈ℕ,
𝖸_i^(N)(·) converges, as N →∞, uniformly on compact sets,
to the continuous process 𝖸_i(·) given by,

    𝖸_i(t)= 𝗑̃_i(t) -x_i-∫_0^t (-η/2𝗑̃_i(s)+γ̃(s)-∑_j=1^∞𝗑̃_j(s)+∑_j=1,j≠ i^∞𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s))ds,   ∀ t ≥ 0.

Observe that, 𝖸(·)=(𝖸_i(·))_i=1^∞ is adapted with respect to (ℱ_t)_t≥ 0. We now claim that for any K ∈ℕ, (𝖸_i(·))_i=1^K is a K-dimensional continuous local martingale with quadratic variation given by,

    ⟨𝖸_i,𝖸_j⟩_t=1_i=j∫_0^t 𝗑̃_i(s)𝗑̃_j(s)ds,   ∀ t ≥ 0,  i,j ∈ℕ.

Let us assume this claim momentarily. Then, we can define, by virtue of the fact that 𝐏̃-a.s. 𝗑̃_i>0, for all i ∈ℕ, the stochastic integral,

    𝗐̃_i(t)def=∫_0^t 1/𝗑̃_i(s)d𝖸_i(s),   ∀ t ≥ 0.

Observe that (𝗐̃_i(·))_i=1^∞ is adapted with respect to (ℱ_t)_t≥ 0 and moreover, for each K ∈ℕ, (𝗐̃_i(·))_i=1^K is a K-dimensional continuous local martingale with quadratic variation,

    ⟨𝗐̃_i,𝗐̃_j⟩_t=1_i=jt,   ∀ t ≥ 0,  i,j ∈ℕ.

Thus, making use of Levy's characterisation theorem <cit.> we get that (𝗐̃_i)_i=1^∞ is a sequence of independent standard Brownian motions. Finally, from (<ref>) we get, for all i∈ℕ,

    𝖸_i(t)=∫_0^t 𝗑̃_i(s)d𝗐̃_i(s),   ∀ t ≥ 0.

This concludes the proof of (<ref>) modulo the claim. We now prove it. Fix K∈ℕ. Define the stopping times ζ_K^N(R) and ζ_K(R), with respect to the corresponding filtrations (ℱ_t^(N))_t≥ 0 and (ℱ_t)_t≥ 0, for R>0, N ≥ K,

    ζ_K^N(R)def=inf{t≥ 0;∑_i=1^K |𝖸_i^(N)(t)|≥ R},   ζ_K(R)def=inf{t≥ 0;∑_i=1^K |𝖸_i(t)|≥ R}.

Observe that, 𝐏̃-a.s., ζ_K(R) R →∞⟶∞. Moreover, arguing similarly to the proof of Theorem <ref> we get that there exists a sequence (R_m)_m=1^∞, with R_m→∞ so that for any fixed m ∈ℕ, ζ_K^N(R_m) ⟶ζ_K(R_m), in distribution, as N →∞. Let t≥ s and R_m be fixed. Let 𝔊 be an arbitrary bounded continuous functional on C([0,s],Ω_+), the space of continuous functions on [0,s] with values in Ω_+. Note that, we have, as N →∞,

    (𝖸^(N)_i(t∧ζ^N_K(R_m))-𝖸^(N)_i(s∧ζ^N_K(R_m)))_i=1^K𝔊[(𝐗̃^(N)(u);0≤ u ≤ s)] 
       d⟶(𝖸_i(t∧ζ_K(R_m))-𝖸_i(s∧ζ_K(R_m)))_i=1^K𝔊[(𝐗̃^Ω_+(u);0≤ u ≤ s)].

Moreover, observe that, if for fixed N ∈ℕ, we define the functional 𝖦 on C([0,s],𝕎_N,+) by,

    𝖦[((f_i(u))_i=1^N;0≤ u ≤ s )] =𝔊[(((N^-1f_i(u))_i=1^∞,N^-1∑_i=1^Nf_i(u)); 0 ≤ u ≤ s)],

with the convention f_i≡ 0, for i>N, then 𝖦 is bounded and continuous. Hence, putting everything together, by virtue of the local martingale property of 𝖸^(N), we obtain,

    𝔼_(𝐱,γ)[(𝖸_i(t∧ζ_K(R_m))-𝖸_i(s∧ζ_K(R_m)))_i=1^K𝔊[(𝐗̃^Ω_+_u;0≤ u ≤ s)]]
       =lim_N →∞𝔼_𝐱^(N)[(𝖸^(N)_i(t∧ζ^N_K(R_m))-𝖸^(N)_i(s∧ζ^N_K(R_m)))_i=1^K𝔊[(𝐗̃^(N)(u);0≤ u ≤ s)]]
       =lim_N →∞𝔼_𝐱^(N)[(𝖸^(N)_i(t∧ζ^N_K(R_m))-𝖸^(N)_i(s∧ζ^N_K(R_m)))_i=1^K𝖦[(𝗑̃^(N)(u);0≤ u ≤ s)]]=(0)_i=1^K.

This proves that (𝖸_i(·))_i=1^K is a continuous local martingale, with (ζ_K(R_m))_m=1^∞ a localising sequence, with respect to (ℱ_t)_t≥ 0. Finally note that, 𝐏̃-a.s. the quadratic variations of the 𝖸^(N)'s also converge, as N →∞, to the desired expression and this concludes the proof of the claim and thus the proof of (<ref>).




We now proceed to show that 𝗑̃ actually solves (<ref>). It will suffice to show that 𝐏̃-a.s., for all t≥ 0, we have  

    ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds= 0.

Observe that, for any M∈ℕ and t≥ 0, we have from (<ref>),
     
    ∑_i=1^M𝗑̃_i(t)=   ∑_i=1^Mx_i+∫_0^t∑_i=1^M𝗑̃_i(s) d𝗐̃_i(s)-η/2∫_0^t∑_i=1^M𝗑̃_i(s)ds+M∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds
       +∫_0^t∑_i=1^M∑_j>M𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

 Therefore, since the first and last terms in the above display are non-negative, one has
     
    M∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds≤∑_i=1^M𝗑̃_i(t)-∫_0^t∑_i=1^M𝗑̃_i(s) d𝗐̃_i(s)+η/2∫_0^t∑_i=1^M𝗑̃_i(s)ds.

  Note that, all the terms on the RHS are finite uniformly as M →∞. Thus, dividing both sides by M, and then, taking M to infinity, one obtains that 𝐏̃-a.s., for all t≥ 0,
 
    ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds≤ 0.

 But clearly, ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds≥ 0 and thus we get 𝐏̃-a.s., ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds= 0, as desired.


 We now want to prove the second statement in the theorem. First observe that, the above implies that 𝐏̃-a.s., for all t≥ 0, 
    γ̃(s)=∑_i=1^∞𝗑̃_i(s),   for almost all  s∈[0,t].

Furthermore, we have, 𝐏̃-a.s., for all M∈ℕ and t≥ 0,
  
    ∑_i=1^M𝗑̃_i(t)=   ∑_i=1^Mx_i+∫_0^t∑_i=1^M𝗑̃_i(s) d𝗐̃_i(s)-η/2∫_0^t∑_i=1^M𝗑̃_i(s)ds+∫_0^t∑_i=1^M∑_j>M𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

Now, by making use of the Burkholder-Davis-Gundy inequality for local martingales, see <cit.>, or directly using Problem 1.5.25 of <cit.>, by virtue of the fact that ∑_i=K^∞𝗑̃_i(·)^2 → 0, uniformly on compact sets, as K →∞, we obtain the existence of a sequence (M_ℓ)_ℓ=1^∞ with M_ℓ→∞, as ℓ→∞, such that 𝐏̃-a.s. the following limit exists uniformly on compact sets,

    𝖴(t)def=lim_ℓ→∞∫_0^t ∑_i=1^M_ℓ𝗑̃_i(s)d𝗐̃_i(s),   ∀ t ≥ 0.

In particular, 𝐏̃-a.s. the function t ↦𝖴(t) is continuous. Then, putting M=M_ℓ and taking the limit ℓ→∞ in both sides of (<ref>), we obtain that 𝐏̃-a.s., for all t≥ 0,
 
    ∑_i=1^∞𝗑̃_i(t)=   ∑_i=1^∞x_i+𝖴(t)-η/2∫_0^t∑_i=1^∞𝗑̃_i(s)ds+lim_ℓ→∞∫_0^t∑_i=1^M_ℓ∑_j>M_ℓ𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

Observe now that the second and third terms on the RHS of (<ref>) are continuous functions of t, and thus, any possible discontinuity of t ↦∑_i=1^∞𝗑̃_i(t) arises from the last term. Note also that since the integrand is strictly positive, one has 
 
    lim_ℓ→∞∫_0^𝗎∑_i=1^M_ℓ∑_j>M_ℓ𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds>lim_t↗𝗎lim_ℓ→∞∫_0^t∑_i=1^M_ℓ∑_j>M_ℓ𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds,

 at any point of discontinuity 𝗎>0 of t ↦∑_i=1^∞𝗑̃_i(t), and hence, from (<ref>) we get 
 
    ∑_i=1^∞𝗑̃_i(𝗎)> lim_t_k↗𝗎∑_i=1^∞𝗑̃_i(t_k)=lim_t_k↗𝗎γ̃(t)=γ̃(𝗎),

where, by virtue of (<ref>), the sequence (t_k)_k=1^∞ is picked so that γ̃(t_k)=∑_i=1^∞𝗑̃_i(t_k) holds for all k∈ℕ and recall that t↦γ̃(t) is continuous.
But, this is a contradiction as we know that
∑_i=1^∞𝗑̃_i(t)≤γ̃(t), for all t≥ 0. Therefore, it follows that
 t ↦∑_i=1^∞𝗑̃_i(t) is, in fact, a continuous function on (0,∞) and in particular, we have that 𝐏̃-a.s.,
 
    ∑_i=1^∞𝗑̃_i(t)=γ̃(t),   ∀ t>0,
 which is equivalent to (<ref>). This completes the proof.
 


 
 Observe that, in the argument above we did not need the full power of convergence in C(ℝ_+,Ω_+) to show convergence of the singular drift term. Nevertheless, one will always need somewhat more information than simply convergence of the paths for this.    
 

 
 It is interesting to note that Theorem <ref> combined with Theorem <ref> give an immediate, alternative, and non-computational, proof of the fact that if, instead of Definition <ref>, we define 𝔐^η as the unique probability measure on Ω_+ satisfying 𝔐^ηΛ^∞_N=𝔐^η_N for all N∈ℕ, then 𝔐^η must be supported on Ω_+^0. The original proof in <cit.> uses the underlying determinantal point process structure and some hard estimates on the correlation kernel of the Laguerre unitary ensemble. The dynamical proof we just mentioned avoids these considerations entirely.
 

We now prove that solutions corresponding to different parameters γ are different in law.
 
  
 In the setting of Theorem <ref>, we denote by (𝐗(t;γ);t≥ 0) the solution to (<ref>) corresponding to γ, then,

    𝖫𝖺𝗐(𝐗(·;γ))≠𝖫𝖺𝗐(𝐗(·;γ̃)), whenever γ≠γ̃.


  
  Based on (<ref>) we have that a.s.

    ∑_i=1^∞𝗑_i(t;γ)
        =γ(t),  ∑_i=1^∞𝗑_i(t;γ̃)
    =γ̃(t),  ∀ t>0.

Note now that γ and γ̃ do not have same distribution, as they are continuous functions with γ(0)≠γ̃(0), and thus, by virtue of (<ref>),

    𝖫𝖺𝗐(∑_i=1^∞𝗑_i(·;γ))≠𝖫𝖺𝗐(∑_i=1^∞𝗑_i(·;γ̃)),

 from which the conclusion follows immediately. 
  
 
 We have the following result which completes the statement of Theorem <ref>.

 
Out of all solutions to the ISDE (<ref>) constructed as in Theorem <ref>, there exists a unique one such that almost surely t ↦∑_i=1^∞𝗑_i(t;γ) is continuous for all t ≥ 0 given by the choice γ=∑_i=1^∞ x_i.
Moreover, this solution is a Markov process. 
 

 We know from Theorem <ref> that 𝐗^Ω_+_·=(𝐗(·;γ),γ(·)) has continuous sample paths. 
 It then follows from continuity of t↦γ(t) and (<ref>) that 
almost surely t ↦∑_i=1^∞𝗑_i(t;γ) is continuous for all t ≥ 0 if and only if γ=∑_i=1^∞ x_i. 


Towards the second assertion, note that by the Markov property of 𝐗^Ω_+, for all 
t≥ s ≥ 0 and all bounded, measurable functions 𝖥:Ω_+→ℝ, we have
 
    𝔼[𝖥(𝐗(t;γ),γ(t))| ℱ_s]=𝔼[𝖥(𝐗(t;γ),γ(t))|σ(𝐗(s;γ),γ(s))],

 where, as before,
(ℱ_t)_t≥ 0
 is the natural filtration of the process (𝐗^Ω_+_t; t≥ 0) and 𝔼[·|σ(𝒞)] denotes conditional expectation with respect to σ(𝒞).

Now, consider the distinguished solution 𝐗(·;γ_*)  of the ISDE with γ_*=∑_i=1^∞ x_i.
Note that, since ∑_i=1^∞𝗑_i(s;γ_*)=γ(s), for all s≥ 0, we have
 
    σ(𝐗(s;γ_*),γ(s))=σ(𝐗(s;γ_*)),  ∀ s≥ 0,

 and thus, in this case the natural filtration (ℱ_t^𝐗(·;γ_*))_t≥ 0 generated by (𝐗(t;γ_*);t≥ 0) is equal to (ℱ_t)_t≥ 0. Denote by 𝒲^∞ the state space of the solution:
 
    𝒲^∞={𝐱∈𝕎_∞,+^∘:∑_i=1^∞ x_i<∞}.

 Now, let 𝖿:𝒲^∞→ℝ be an arbitrary bounded measurable function and take 𝖥 to be its obvious lift on Ω_+. Thus summarising, by virtue (<ref>) and (<ref>), we have
  
    𝔼[𝖿(𝐗(t;γ_*))|ℱ_s^𝐗(·;γ_*)]=𝔼[𝖿(𝐗(t;γ_*))|σ(𝐗(s;γ_*))],  ∀ s≥ 0,

 which implies that  (𝐗(t;γ_*))_t≥ 0 is a Markov process and concludes the proof. 
 


    This follows from Theorem <ref>, Theorem <ref>, Theorem <ref>, and Proposition <ref>.





    By Theorem <ref>, the probability measure 𝔪 corresponds to the unique sequence of probability measures (𝔪_N)_N=1^∞ on (𝕎_N,+)_N=1^∞ given by,
    
    𝔪_N=𝔪Λ_N^∞,   ∀ N∈ℕ.

    Let 𝗑^(N) be the solution of (<ref>) started at 𝗑^(N)(0)d=𝔪_N. By combining Theorem <ref> and Lemma <ref>, a standard argument gives that, under the embedding (<ref>) of 𝕎_N,+ into Ω_+, 𝔪_N converges weakly to 𝔪. Namely, 𝐗^(N)_0d⟶𝐗^Ω_+_0, and hence,
 𝐗^(N)d⟶𝐗^Ω_+ by Theorem <ref>. One can then follow the argument of Theorem <ref> to prove that (𝐗(t;𝔪);t≥ 0) is a weak solution of the ISDE (<ref>) with initial condition 𝐗(0;𝔪)d=𝖨𝖡𝖾𝗌_η. Note that, as 𝔐^η is the invariant measure of 𝐗^Ω_+, we have 
 
    𝖫𝖺𝗐(𝐗(t;𝔐^η))=𝖨𝖡𝖾𝗌_η,   ∀ t ≥ 0,

 On the other hand, if 𝔪≠𝔐^η, we must have
 
    ℙ_𝔪(γ(0;𝔪)>∑_i=1^∞𝗑_i(0;𝔪))>0,

 where ℙ_𝔪 denotes 𝖫𝖺𝗐(𝐗^Ω_+) if 𝖫𝖺𝗐(𝐗_0^Ω_+)=𝔪. Thus, based on the arguments in Theorem <ref>, since γ(t;𝔪)=∑_i=1^∞𝗑_i(t;𝔪) for all t>0, and recall that t↦γ(t;𝔪) is continuous, we obtain that the stochastic processes t ↦∑_i=1^∞𝗑_i(t;𝔪) and t↦∑_i=1^∞𝗑_i(t;𝔐^η) are not equal in law, for any 𝔪≠𝔐^η, and so, 

    𝖫𝖺𝗐(𝐗(·;𝔐^η))≠𝖫𝖺𝗐(𝐗(·;𝔪)).
 
That is, the following holds if and only if 𝔪=𝔐^η, 
 
    𝖫𝖺𝗐(𝐗(t;𝔪))=𝖨𝖡𝖾𝗌_η,   ∀ t ≥ 0,

and this completes the proof.

 












§ DYNAMICAL CAUCHY MODEL



In this section, we consider stochastic dynamics related to the Cauchy <cit.> or so-called Hua-Pickrell ensemble <cit.>. This is the following probability measure on 𝕎_N,
depending on a parameter s∈ℂ, with (s)>-1/2,

    𝔑_N^(s)(d𝐱)=1/𝒵_N^(s)Δ_N^2(𝐱)
        ∏_j=1^N(1 + x_j^2)^-(s)-Ne^2(s)tan^-1(x_j)d𝐱,
 
where,
𝒵_N^(s) is a normalizing constant.

The associated N-particle dynamics, that we call the dynamical Cauchy model, introduced in <cit.> (they were Hua-Pickrell diffusions therein),  is given by the unique solution to the following system of stochastic equations
 
    d𝔵_i(t) = √(2[𝔵_i(t)^2+1])d𝗐_i(t) + 2[(1-N-(s))𝔵_i(t)+(s)]dt+2∑_j=1:j≠ i^N𝔵_i(t)^2+1/𝔵_i(t)-𝔵_j(t)dt,
 
for i=1,…, N, where as usual the 𝗐_i are independent standard Brownian motions.
It is proven in <cit.>, making use of the results of <cit.>,  that
(<ref>) has a unique strong solution for any initial condition 𝐱∈𝕎_N and almost surely for all t>0, 𝗑(t)∈𝕎_N^∘ and moreover the associated semigroup (𝔔^(s)_N(t))_t≥ 0 is Feller. The following result is the main theorem of <cit.>.


Let s∈ℂ. Then, the semigroups (𝔔_N^(s)(t))_t≥ 0 are intertwined:

    𝔔_N+1^(s)(t)Λ_N^N+1=
    Λ_N^N+1𝔔_N^(s)(t),  ∀ t≥ 0,    N∈ℕ.

In particular, there exists a unique Feller semigroup (𝔔_∞^(s)(t))_t≥ 0, with associated stochastic process (𝐗_t^Ω;t≥ 0) on Ω, defined via the intertwining

    𝔔_∞^(s)(t)Λ_N^∞=
    Λ_N^∞𝔔_N^(s)(t),  ∀ t≥ 0,    N∈ℕ.
   
Moreover, if (s)>-1/2 then (𝔔_∞^(s)(t))_t≥ 0 has a unique invariant measure 𝔑_∞^(s) uniquely determined via the relation

    𝔑_∞^(s)Λ_N^∞ = 𝔑_N^(s),   ∀ N ∈ℕ.




Before giving our main result on the dynamical Cauchy model it is worth pointing out, that for real s>-1/2, the invariant measure associated to the dynamics (𝔔^s_∞(t))_t≥ 0 is known to admit an explicit description. We briefly spell this out.


Let s>-1/2. The Hua-Pickrell point process 𝖧𝖯^Conf_s with parameter s is the determinantal point process on (-∞,0)×(0,∞) with correlation kernel 𝔎_s given by 
    
    𝔎_s(z,w) = 1/2πΓ(s+1)^2/Γ(2s+1)Γ(2s+2)𝒴_1^(s)(z)𝒴_2^(s)(w)-𝒴_1^(s)(w)𝒴_2^(s)(z)/z-w,

where the functions 𝒴_1^(s)(z), 𝒴_2^(s)(z) are given by 

    𝒴_1^(s)(z) = 2^2s-1/2Γ(s+1/2) ·1/|z|^1/2J_s-1/2(1/|z|),   𝒴_2^(s)(z) = 2^2s+1/2Γ(s+3/2) ·1/|z|^1/2 J_s+1/2(1/|z|),

where as before J_r denotes the Bessel function with parameter r.


For s=0 and under the transformation x↦ x^-1, 𝖧𝖯^Conf_0 becomes the sine determinantal point process, the universal limit of random matrices in the bulk of the spectrum <cit.>. Since 𝖧𝖯^Conf_s consists of distinct points it gives rise to a unique probability measure  𝖧𝖯_s on 𝕎_∞,+^∘×𝕎_∞,+ by labelling its positive and negative points respectively according to their absolute values. Define the space

    Ω^0def={ω=(𝐱^+,𝐱^-,γ,δ)∈Ω:γ=lim_k→∞∑_i=1^∞[x_i^+1_x_i^+≥ k^-2- x_i^-1_x_i^-≥ k^-2] , δ=∑_i=1^∞[(x^+_i)^2+(x^-_i)^2]},

and write π:Ω→𝕎_∞,+×𝕎_∞,+ for the map π((𝐱^+,𝐱^-,γ,δ))=(𝐱^+,𝐱^-). Then, it turns out that 𝔑_∞^(s) is the unique probability measure on Ω which is supported on Ω^0 and moreover satisfies (see the discussion in <cit.> and the references therein),

    π_*𝔑^(s)_∞=𝖧𝖯_s.





We now upgrade the abstract construction of (𝐗^Ω_t;t≥ 0) from Theorem <ref> to a strong approximation theorem from the finite dimensional dynamics which moreover implies that (𝐗^Ω_t;t≥ 0) has continuous sample paths. Finally, we prove convergence to equilibrium from any initial condition. We leave the very interesting problem of constructing an ISDE, the drift of which will most likely require renormalisation to even make sense, to future work. We define 𝐗^(N)=((𝗑^(N),+_i)_i=1^∞,(𝗑_i^(N),-)_i=1^∞,γ^(N),δ^(N)) to be the process on Ω obtained from (<ref>) under the embedding (<ref>) of 𝕎_N into Ω.





Let s∈ℂ. Consider the Feller process (𝐗_t^Ω;t≥ 0) with semigroup (𝔔_∞^(s)(t))_t≥ 0 constructed in Theorem <ref> and the embedded processes 𝐗^(N) on Ω defined above. Let 𝐗_0^Ω=ω∈Ω be arbitrary and assume 𝐗^(N)(0)→ω in the topology of Ω. Then, as N →∞,

    𝐗^(N)d⟶𝐗^Ω in  C(ℝ_+,Ω).

Moreover, there exists a coupling of the 𝐗^(N) and 𝐗^Ω such that, almost surely, for all T≥0, we have:
    
    sup_t∈[0,T](∑_i=1^∞|𝗑_i^(N),+(t)-𝗑_i^+(t)|^3+∑_i=1^∞|𝗑_i^(N),-(t)-𝗑_i^-(t)|^3)N→∞⟶ 0.

   Finally, for (s)>-1/2 and 𝔎 an arbitrary probability measure on Ω, if 𝖫𝖺𝗐(𝐗_0^Ω)=𝔎, then as t→∞,

    𝐗_t^Ωd⟶𝐙,   where 𝖫𝖺𝗐(𝐙)=𝔑_∞^s.




All the statements in the theorem, except for convergence to equilibrium, follow by combining Theorem <ref> and the fact that C^∞_c,sym(𝕎_N) is a core, and invariant under the generator of  (𝔔_N^(s)(t))_t≥ 0 (this can be proven in exactly the same way as Proposition <ref>), by virtue of Theorem <ref> and Proposition <ref>. The only thing that remains to note is convergence to equilibrium of the finite-dimensional processes which itself follows from that of the corresponding matrix process shown in <cit.>. Then, the remaining statement in the theorem follows from Theorem <ref>.



acm




School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.mailto:theo.assiotis@ed.ac.uktheo.assiotis@ed.ac.uk



School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.mailto:theo.assiotis@ed.ac.ukZ.S.Mirsajjadi@sms.ed.ac.uk


