Paper Title: Finite rank perturbation of non-Hermitian random matrices: heavy tail and sparse regimes.





§ ABSTRACT 
We revisit the problem of perturbing a large, i.i.d. random matrix by a finite rank error. It is known that when elements of the i.i.d. matrix have finite fourth moment, then the outlier eigenvalues of the perturbed matrix are close to the outlier eigenvalues of the error, as long as the perturbation is relatively small. We first prove that under a merely second moment condition, for a large class of perturbation matrix with bounded rank and bounded operator norm, the outlier eigenvalues of perturbed matrix still converge to that of the perturbation. We then prove that for a matrix with i.i.d. Bernoulli (d/n) entries or Bernoulli (d_n/n) entries with d_n=n^o(1), the same result holds for perturbation matrices with a bounded number of nonzero elements. 



Infinite dimensional metapopulation SIS model with generalized incidence rate
    Pierre-André Zitt
    August 5, 2024
=============================================================================





§ INTRODUCTION


Consider A_n:={a_ij}_1≤ i,j≤ n an n by n random matrix with i.i.d. elements a_ij having mean zero and finite second moment: 
    𝔼[a_ij]=0  ,and 𝔼[|a_ij|^2]=1.
 When a_ij's have complex Gaussian distribution, then A_n belongs to the complex Ginibre ensemble <cit.>. For A_n with i.i.d. entries with mean 0, variance one and otherwise arbitrary distribution, it is known (<cit.>, culminating many previous works, see references therein) that the empirical spectral density of n^-1/2A_n converges to the circular law, i.e. the uniform distribution in the unit disk in the complex plane. 

Determining the outlier eigenvalues of A_n is a very different task. When entries a_ij have finite fourth moments, then the operator norm of n^-1/2A_n is bounded thanks to a standard truncation procedure (see <cit.> <cit.>) so the modulus of its largest eigenvalue
converges to 1 almost surely. When a_ij has a finite second moment but infinite fourth moment, then the operator norm of n^-1/2A_n tends to infinity, but it was later uncovered that under merely assumption <ref>, the modulus of the largest eigenvalue of n^-1/2A_n still converges to 1 as n tends to infinity, i.e. no outlier exists for the circular law. This no-outlier result was first proven via a moments method in <cit.> under additional conditions, and then proven in full generality via the characteristic functions in <cit.>. 

Another model of interest concerns adjacency matrices of sparse Erdős–Rényi digraphs, that is, we consider A_n={a_ij}_1≤ i,j≤ n with each a_ij i.i.d. having distribution Ber(d_n/n),d_n>1. For these sparse matrices, we know that (<cit.>,<cit.>) there is only one outlier eigenvalue of (d_n)^-1/2A_n (tending to infinity when d_n→∞ or converging to √(d) for d_n=d), and all other eigenvalues have modulus 1+o(1). The circular law for (d_n)^-1/2A_n in the regime d_n→∞ was recently proved in full generality in <cit.>, and the existence of limiting spectral density in the case of constant d was recently resolved in <cit.>.

A closely related stream of research concerns finite rank perturbations of random matrices, that is, what are the outlier eigenvalues of n^-1/2A_n+C_n, where C_n is a matrix with finite rank and bounded operator norm? The corresponding problem for Hermitian random matrices has been studied in more detail in <cit.>, <cit.>, <cit.>, <cit.>, <cit.> (see also references therein): there is a phase transition called the BBP transform: for a Wigner matrix W_n, a unit vector v and θ>0, the largest eigenvalues of n^-1/2W_n+θ vv^t converges to θ+1/θ when θ>1 and converges to 2 when θ<1. In the non-Hermitian case of A_n with i.i.d. elements, the outlier eigenvalues of n^-1/2A_n+C_n behave rather differently: 
it is first proved in Tao <cit.> that in this i.i.d. case, under a fourth moment assumption on matrix entries, the outlier eigenvalues of n^-1/2A_n+C_n converge towards the eigenvalues of C_n with modulus larger than one, so that the mapping θ→θ+1/θ in the Wigner case is replaced by the identity mapping in this i.i.d. case when |θ|>1.  Fluctuations around the limiting value were then studied in <cit.>, and see also <cit.> for outliers in a related case: the single ring theorem. Meanwhile, finite rank perturbations for elliptic random matrices were studied in <cit.> also under a finite fourth moment condition, and the result interpolates between the i.i.d. case and the Wigner case, giving a mapping θ→θ+ρ/θ for |θ|>1, where ρ∈(-1,1) is the covariance profile of the matrix, i.e. 𝔼[a_ija_ji]=ρ.

The main objective of this paper is to investigate the effect of infinite fourth moment and sparsity on the outlier eigenvalues of n^-1/2A_n+C_n. Technically, a lot of complications arise in this setting: the operator norm of n^-1/2A_n is unbounded, so that arguments based on operator norms, as in <cit.>, do not apply. We also cannot use the replacement principle to compare A_n to a matrix with Gaussian entries as in <cit.> as the replacement principle typically relies on a fourth moment assumption. Likewise,we find it difficult in the computations to use singular value decomposition and simplify C_n into a canonical form, because the operator norm of n^-1/2A_n is not bounded. We have to turn back to first principles. Inspired by <cit.>, we use the characteristic function method to find outlier eigenvalues. 

The main results of this paper are listed as follows. We use the notation H(𝔻) to denote the space of holomorphic functions on the unit disk 𝔻, equipped with the compact open topology, i.e. the topology of uniform convergence on compact subsets. A sequence c_n∈ H(𝔻) is said to form a tight sequence if for any ϵ>0 there is a compact subset K_ϵ⊂ H(𝔻) such that ℙ(c_n∈ K_ϵ)≥ 1-ϵ for each n.

 Assume that C_n=(C_ij)_1≤ i,j≤ n is a deterministic n× n square matrix that belongs to one of the following two classes:

    
  * C_n is a diagonal matrix of finite rank, or C_n is in a Jordan normal form with finite rank, or more generally C_n has only M_1 non-zero entries. In all these cases, assume moreover that all the elements of C_n are bounded in norm by M_2, and constants M_1,M_2 are independent of n,
    
  * C_ij=μ/n for each i,j, or more generally |C_ij|≤M/n for some M>0 and each i,j, where M is independent of n.

Assume also that the following condition holds in each previous case: denote by c_n(z) the reverse characteristic function of C_n: c_n(z)=(I-zC_n), then c_n(z) forms a tight family in H(𝔻), and there exists some non-zero function c(z)∈H(𝔻) such that c_n(z) converges to c(z) in H(𝔻) as n tends to infinity.


Assume also that A_n=(a_ij)_1≤ i,j≤ n is a square matrix of size n, with independent, identically distributed entries a_ij satisfying 
    𝔼[a_ij]=0  ,and 𝔼[|a_ij|^2]=1.

(In particular, a_ij may not have a finite fourth moment.)


    Then almost surely, as n tends to infinity, the eigenvalues of n^-1/2A_n+C_n with modulus greater than 1+o(1) converge (after relabeling) to the solutions z of c(z^-1)=0 with |z|>1, and every solution to c(z^-1)=0 with |z|>1 arises as the limit of some eigenvalues of n^-1/2A_n+C_n.

    Along the proof we will also prove the following convergence of characteristic functions: denote by q_n(z)=(I-z(n^-1/2A_n+C_n)), then q_n(z) converges in distribution to q(z) as functions in H(𝔻), where 
    
    q(z)=κ(z)c(z)e^-F(z),   z∈𝔻,
 
    where κ(z)=√(1-𝔼[a_11^2]z^2), F(z)=∑_k=1^∞ X_kz^k/k where {X_k}_k≥ 1 are independent complex Gaussian variables with 

    𝔼[X_k]=0,  𝔼[|X_k|^2]=1,  𝔼[X_k^2]=𝔼[a_11^2]^k.



As a special case, when the finitely many eigenvalues of C_n are n-independent, then our assumption that c_n(z) converges to c(z) is always guaranteed. 

Our assumptions on C_n are a bit technical: in practice, one should be able to check that the result of Theorem <ref> is true for any C_n with bounded rank and bounded operator norm. The combinatorics involved in the computation for general C_n is however very complicated, as we cannot use directly the singular value decomposition for C_n (if n^-1/2A_n has bounded operator norm, then SVD can be effectively used as we can ignore the unitary matrices in SVD and reduce C_n to a normal form, but this is not the case here with only two finite moments). Under a finite fourth moment condition, this theorem (in more general form) was proved in <cit.>, see also <cit.>.

We also note that the constants M_1,M_2 and M in the statement of Theorem <ref> can be slowly growing in n at rate O(n^o(1)) with exactly the same proof.

The main achievement of this result compared to previous works is that we do not assume entries a_ij have a finite fourth moment, and under merely finite second moment, we have the same finite-rank perturbation result. Note that our proof does not yield the fluctuations of outlier eigenvalues around their limiting value, which is studied in <cit.> for some special cases of C_n under finite fourth moment assumption. Also, for (symmetric) Wigner matrices with entries having less than four moments, its finite rank perturbation result is recently obtained in <cit.> and is completely different from the non-Hermitian case here.

For symmetric (or Hermitian) random matrices, Van Handel and Brailovskaya <cit.> recently proposed a general framework, further developing the ideas in <cit.>, that can prove finite rank perturbation results for random matrices only assuming very few moments. The main motivating example in <cit.> can be stated as follows: consider W_n=(W_ij) a real symmetric random matrix with (W_ij)_1≤ i≤ j≤ n independent, that W_n has mean 0 and has the same variance profile as a GOE matrix. Then assuming moreover that n^-1/2|W_ij|≤ (log n)^-2 a.s. for each i,j, the finite rank perturbation properties of W_n are asymptotically the same as a GOE matrix: we still have the phase change at θ=1 and the mapping θ→θ+1/θ when |θ|>1, (the fluctuation around the spiked eigenvalues are different though). This result is remarkable in that only a second moment condition is assumed, but the condition n^-1/2|W_ij|≤ (log n)^-2 is necessary: without it, the top eigenvalues are Poisson distributed, see <cit.>. The main result of this paper also assumes only a finite second moment, yet we do not impose any truncation condition and still get the same finite rank perturbation as the subGaussian case: this is because the i.i.d. nature of our matrix leads to more cancellations than Wigner matrices. For this reason, we reveal a picture very different from that in <cit.>. Also, <cit.> uses universality of resolvent (or Stieltjes transform) which is very convenient when eigenvalues are on the real line, but much more work is needed when eigenvalues are dispersed on the complex plane. 

In a forthcoming work <cit.>, the author further uses free probability ideas from <cit.> to study finite rank perturbations for non-Hermitian random matrices. The results presented in <cit.> have a different flavour from those presented here: in <cit.>, we consider random band matrices that either have independent entries or have an elliptic covariance profile, whereas band matrices are outside the scope of techniques of this paper. On the other hand, the techniques in <cit.> use a Hermitization procedure, and therefore excludes the case with only two bounded moments and very sparse random graphs, as considered here. It seems very hard to find a theory that unifies both the method of characteristic functions as in this paper, and the method of free probability and concentration inequalities in <cit.>. We hope to find a general powerful method that unites both results in <cit.> and in this paper.




 Another important case consists of i.i.d matrix with non-zero mean, that is, we consider C_ij=μ/√(n). This case is however not covered by the techniques of this paper, as in this case the characteristic function of n^-1/2A_n+C_n is no longer a perturbation of that of n^-1/2A_n, and it is not clear if the method of characteristic functions still applies. When a_ij have finite fourth moment, it is proved in <cit.> that except the single outlier eigenvalue at μ√(n)+o(1), all other eigenvalues lie in {z∈ℂ:|z|≤ 1+o(1)}. The author is not sure if exactly the same result can be proved under only a second moment condition, and this is left for future work. The existence of exceptional eigenvalue to this matrix model with non-zero mean was first noticed in <cit.>, and more precise information on the distribution of this eigenvalue can be found in <cit.> under additional moment conditions.





 §.§ Sparse i.i.d. matrices: finite rank perturbation

The second major result of this paper concerns sparse non-Hermitian random matrices. To our best knowledge, finite rank perturbations of these sparse i.i.d. matrices have not been studied before in the literature.




  * Let A_n be a n× n random matrix with i.i.d. Bernoulli (d/n) entries (d>1). Let C_n=(C_ij)_1≤ i,j≤ n be a deterministic n× n matrix with each entry bounded by M_1 and at most M_2 non-zero elements (M_1,M_2 are independent of n). Denote by c_n(z)=(I-zC_n) the reverse characteristic function and assume that c_n(z) converges to c(z) as holomorphic functions on {z∈ℂ:|z|<1/√(d)}. 

Then almost surely, as n tends to infinity, all eigenvalues of A_n+C_n lie in {z∈ℂ:|z|≤√(d)(1+o(1))} except the following outliers: an outlier eigenvalue converges to d with multiplicity one, and all the other outliers converge to the solutions of c(z^-1)=0 with |z|>√(d). Meanwhile, each solution to c(z^-1)=0 with |z|>√(d) correspond to the limit of a sequence of outlier eigenvalues of A_n+C_n.


  * Let A_n be a n× n random matrix with i.i.d. Bernoulli (d_n/n) entries where d_n→∞ as n tends to infinity and d_n=n^o(1). Let C_n=(C_ij)_1≤ i,j≤ n be a deterministic n× n matrix with each entry bounded by M_1 and at most M_2 non-zero elements (M_1 and M_2 are independent of n). Denote by c_n(z)=(I-zC_n) and assume that c_n(z) converges to c(z) as holomorphic functions on 𝔻. 

Then almost surely, as n tends to infinity, all eigenvalues of (d_n)^-1/2A_n+C_n lie in {z∈ℂ:|z|≤ 1+o(1)} except the following outliers: one outlier eigenvalue tends almost surely to infinity, and all the other outlier eigenvalues converge to a solution to c(z^-1)=0 with |z|>1. Meanwhile, each solution to c(z^-1)=0 with |z|>1 correspond to the limit of a sequence of outlier eigenvalues of (d_n)^-1/2A_n+C_n.




We have also proved that the reverse characteristic function (I_n-z((d_n)^-1/2A_n+C_n)) converges to some limiting random analytic functions, and the limit is closely related to random analytic functions derived in <cit.>.



    Under the assumptions in case (2), it is proved in <cit.>, <cit.> that the empirical spectral density of A_n converges to the circular law. Very recently, existence of limiting empirical spectral density of A_n under the assumption of case (1) was proved in <cit.>.





We have ruled out the case where C_n has many non-zero elements, say for example C_i,j=μ/n for some μ≠ 0.  In this case the rank one perturbation result should be different from the stated result: we expect that in case (1), the outlying eigenvalue may not converge to d, but to some other limit. The proof of such claims may involve complicated computations, which we choose to omit for sake of simplicity.




§ PROOF IN THE HEAVY-TAILED CASE




We begin with some very general reduction techniques that apply to any deterministic C_n satisfying the assumption of Theorem <ref>. In this case the characteristic function has the following expansion:

    q_n(z)=(1-z(A_n/√(n)+C_n))=1+∑_k=1^n (-z)^kP_k^n

where 


    p_k^n=∑_I⊂{1,2,⋯,n}:|I|=kn^-k/2((A_n+√(n)C_n)(I))

with (A_n+√(n)C_n)(I)={a_jk+√(n)c_jk}_j,k∈ I.


The key point here (adopted from <cit.>) is that, once we can show q_n(z) is tight as an element in H(𝔻), then the deduction of convergence of q_n(z) can be obtained from the convergence of individual coefficients p_k^n(z). We note that it seems easier to check the convergence of individual coefficients P_k^n and the tightness of q_n, instead of proving directly that q_n converges in H(𝔻). The latter claim follows from the following lemma, see <cit.>, Lemma 3.2:
(<cit.>, Lemma 3.2)
Given f_n(z)=∑_k=0^∞ z_kf_k^n a sequence of power series, and assume that {f_n(z)} are tight in H(𝔻). Then suppose that for any m≥ 0,

    (f_0^n,⋯,f_m^n)lawn→∞⟶
     (f_0,⋯,f_m),

then f=∑_k=0^∞ z^k f_k is defined in H(𝔻) and f_nlawn→∞⟶ f.


A further technical reduction is truncation to the case where all the a_ij are bounded. Given M>0 define 
    A_n^M={a_ij^M}_1≤ i,j≤ n,   a_ij^M=a_ij1_|a_ij|≤ M-𝔼[a_ij1_|a_ij|≤ M].

Then defining 
    p_k^n,M=∑_I⊂{1,2⋯ n}:|I|=kn^-k/2((A_n^M+n^1/2C_n)(I)),
 we will need the following lemma:


Suppose that there exists a sequence of coefficients (Y_1^M,⋯,Y_k^M) such that 

    (P_1^n,M,⋯,P_k^n,M)
    lawn→∞⟶ (Y_1^M⋯,Y_k^M),   and  (Y_1^M⋯,Y_k^M)lawM→∞⟶ (Y_1,⋯,Y_k),
for all M>0, then we have 

    (P_1^n,⋯,P_k^n)lawn→∞⟶ (Y_1,⋯,Y_k).


This lemma in the C_n=0 case is already proved in <cit.>, Lemma 3.3. The proof in the general case is exactly the same as C_n is just a deterministic matrix with bounded entries. The proof is left to the reader.

Now we use another reduction, which is already observed in <cit.>: at least when z is sufficiently small, we have that 

    B_n:=-∑_k=1^∞ (A_n+√(n)C_n/√(n))^kz^k/k

is well-defined and converges to the logarithm of 1-zA_n+√(n)C_n/√(n). From the matrix relation (e^B_n)=e^TrB_n, this leads to, for z sufficiently small, 

    q_n(z)=exp(-∑_k=1^∞Tr((A_n+√(n)C_n)^k)/n^k/2z^k/k).
 Since two analytic functions that are equal in a small neighborhood of complex plane must be equal identically, this leads to the following consequence:

For each k∈ℕ_+, the coefficients (P_1^n,⋯,P_k^n) can be represented as (n-independent) polynomial functions of 
    (Tr((A_n+√(n)C_n))/√(n),⋯,Tr((A_n+√(n)C_n)^k)/n^k/2)
 (and in reverse). Thanks to this expression, all the further effort of proving convergence of q_n(z) boils down to proving the convergence of Tr(A_n+√(n)C_n)^k/n^k/2 as n tends to infinity.




 §.§ Illustrative example: a diagonal matrix of rank one



For illustrative purpose, we begin with the proof of Theorem <ref> in the simplest case where C_n is a diagonal matrix of rank one. Then without loss of generality we may assume that C_n=θ E_1,1 for some θ≠ 0, where E_11 is the n by n matrix with 1 in its entry (1,1) and 0 everywhere else. We expand the trace as follows:

    Tr((A_n+√(n)C_n)^k)   =∑_1≤ i_1,i_2,⋯, i_k≤ n,
    |{i_1,⋯,i_k}|=ka_i_1i_2a_i_2i_3⋯ a_i_ki_1+∑_1≤ i_1,i_2,⋯ i_k≤ n,
     |{i_1,⋯,i_k}|≤ k-1 a_i_1i_2a_i_2i_3⋯ a_i_ki_1
       +∑_ℓ=1^k-1(√(n)θ)^ℓ∑_1≤ i_1,i_2,⋯, i_k-ℓ≤ n,
    |{i_1,⋯,i_k-ℓ}|=k-ℓ,
     1∈{i_1,⋯,i_k-ℓ}a_i_1i_2a_i_2i_3⋯ a_i_k-ℓ i_1
       +∑_ℓ=1^k-1(√(n)θ)^ℓ∑_1≤ i_1,i_2,⋯, i_k-ℓ≤ n,
    |{i_1,⋯,i_k-ℓ}|<k-ℓ,
     1∈{i_1,⋯,i_k-ℓ}a_i_1i_2a_i_2i_3⋯ a_i_k-ℓ i_1
       +(√(n)θ)^k.


Note that thanks to Lemma <ref>, we may assume that a_ij are bounded and |a_ij|≤ M almost surely.

Let ℭ_k^n be the set of k-directed cycles in {1,2,⋯,n}, where a k-directed cycle is defined as a directed cycle of length k on indices 1,2,⋯,n having k different vertices. Given g∈ℭ_k^n denote by a_g=∏_e  edge of g a_e. Then the summation

    ∑_1≤ i_1,i_2,⋯, i_k≤ n,
    |{i_1,⋯,i_k}|=ka_i_1i_2a_i_2i_3⋯ a_i_ki_1

can be restated as

    t_k^n=∑_g∈ℭ_k^n a_g.


The joint distribution of t_k^n can be computed as in <cit.>, Lemma 3.4:

(<cit.>, Lemma 3.4)
    Given any k_1,⋯,k_m≥ 1 and any sequence s_1,⋯,s_m∈{·,*}:
    
    𝔼[(t_k_1^n/n^k_1/2)^s_1⋯(t_k_m^n/n^k_m/2)^s_m]→𝔼[(X_k_1/√(k_1))^s_1⋯(X_k_m/√(k_m))^s_m],
 with the convention that x^·=x and x^*=x̅,
    and that {X_k}_k≥ 1 are a sequence of independent complex Gaussian random variables with 𝔼[X_k]=0, 𝔼[|X_k|^2]=1 and 𝔼[X_k^2]=𝔼[a_11^2]^k for any k≥ 1.


Now we analyze the term on the second line of (<ref>). Let ℭ_k-ℓ,1^n be the set of k-ℓ-directed cycles in {1,2,⋯,n} that contain 1. Given g∈ℭ_k-ℓ^n denote by a_g=∏_e  edge of g a_e. Then for the second line of (<ref>), it suffices to study, for each ℓ=1,⋯,k: 

    t_k,1^ℓ,n=(√(n)θ)^ℓ∑_g∈ℭ_k-ℓ,1^n a_g.


We will now state a series of lemmas and prove them in the sequel. We begin with


    For each k and each ℓ=1,2,⋯,k-1, we have the convergence in probability
    
    t_k,1^ℓ,n/n^k/2lawn→∞⟶ 0.





Having determined the limiting random term, at this time it suffices to consider the limiting constant term


    r_k^n=∑_1≤ i_1,⋯,i_k≤ n
     |{i_1,⋯,i_k}|<ka_i_1i_2a_i_2i_3⋯ a_i_k-1i_ka_i_ki_1.

and for each ℓ=1,⋯,k-1, define

    r_k,1^ℓ,n=(√(n)θ)^ℓ∑_1≤ i_1,i_2,⋯, i_k-ℓ≤ n,
    |{i_1,⋯,i_k-ℓ}|≤ k-ℓ-1,
    1∈{i_1,⋯,i_k-ℓ}a_i_1i_2a_i_2i_3⋯ a_i_k-ℓi_1.

For r_k^n, we have the following deterministic limit
from <cit.>: 
(<cit.>, Lemma 3.5) We have the following deterministic limit

    r_k^n/n^k/2lawn→∞⟶𝔼[a_11^2]^k/2,    k  even ,
        
     0,    k  odd .




For r_k,1^ℓ,n we will also prove a similar result:
 We have the following convergence for each ℓ=1,⋯,k-1:
    
    r_k,1^ℓ,n/n^k/2lawn→∞⟶    0.




 It suffices to expand

    n^-k𝔼[|t_k,1^ℓ,n|^2]
 and compute the expectation. For multi-indices 𝐢=(i_1,⋯,i_k-ℓ), 𝐣=(j_1,⋯,j_k-ℓ) in ℭ_k-ℓ,1^n, define a_𝐢=a_i_1i_2a_i_2i_3⋯ a_i_k-ℓi_i . Then in order for 𝔼[a_𝐢a_𝐣]≠ 0, one must have that the union of the edges traversed by 𝐢 and 𝐣 must each be traversed at least twice, because each a_ij has mean zero. 

If there is an edge traversed more than twice by the union 𝐢∪𝐣, then by the handshaking lemma, the union of indices contained in 𝐢∪𝐣 is less than k-ℓ, and except the index 1 already fixed, there are strictly less than k-ℓ-1 number of distinct indices, and the contribution to the overall expectation (<ref>) of these terms is less than O(n^-kn^k-ℓ-2n^ℓθ^2ℓ(4M)^2k)=O(n^-2) (where M is the almost sure bound on a_ij thanks to Lemma <ref>) and hence this term is negligible.

In the remaining case, each edge in the union of 𝐢∪𝐣 (multi-edges counted multiple times) are traversed exactly twice, and hence via a graphical reasoning, 𝐢∪𝐣 must be a doubled circle with k-ℓ edges and at most k-ℓ-1 indices different from 1. The contribution of these terms to the expression (<ref>) is at most 

    n^-k n^k-ℓ-1n^ℓθ^2ℓ(4M)^2k=O(n^-1),
 and hence the lemma is proved.





    We first prove that 
     
    𝔼[ r_k,1^ℓ,n/n^k/2]n→∞⟶     0.

     To see this, observe that for a cycle to contribute to the sum in expectation, from the cycle (i_1,i_2,⋯,i_k-ℓ) we construct a multigraph on i_1,⋯,i_k-ℓ and every edge has to be repeated twice for it to contribute to the expectation. Next, if some edge has outer degree three, the contribution to the limit of such graphs is negligible by a similar reasoning as in previous lemma. Thus for the sum to contribute, the multigraph must be a multicircle and k-ℓ has to be even. Regardless of the parity of k-ℓ, the number of circles of length k-ℓ/2 that has 1 in its indices is at most n^k-ℓ/2-1. Thus |𝔼[r_k,1^ℓ,n]|≤ n^k/2-1 and hence the claim is proved. 

We next prove that Var[r_k,1^ℓ,n/n^k/2]n→∞⟶    0. We follow similar steps as in the proof of <cit.>,Lemma 3.5. For any 𝐢=(i_1,⋯,i_k-ℓ), denote by a_𝐢=a_i_1i_2a_i_2i_3⋯ a_i_k-ℓa_i_i, we have trivially the bound 

    𝔼[(a_𝐢-𝔼(a_𝐢))(a_𝐣-𝔼(a_𝐣))]≤ 4M^2k
 for any constant M larger than the modulus of the support of a_11. 

Meanwhile, for any index 𝐢, we regard 𝐢 as a path of length k and define a graph (V_𝐢,E_𝐢) where V_𝐢={i_1,⋯,i_k-ℓ} and E_𝐢={(i_r,i_r+1), r=1,⋯,k-ℓ}. 

Define the excess of graph (V_𝐢,E_𝐢) as:

    χ_𝐢=Card(E_𝐢)-Card(V_𝐢)+1≥ 0,
 which is the maximal number of edges to be removed from (V_𝐢,E_𝐢) so that the resulting graph has no undirected cycles.
Then by definition χ_𝐢≥ 2. Consider another multi-index 𝐣 satisfying the same assumption, then χ_𝐣≥ 2. We join the graphs (V_𝐢,E_𝐢) and (V_𝐣,E_𝐣) as follows: 

    V_𝐢𝐣=V_𝐢∪ V_𝐣,   E_𝐢𝐣=E_𝐢∪ E_𝐣.

Then defining χ_𝐢𝐣=Card(E_𝐢𝐣)-Card(V_𝐢𝐣)+c with c=1 when V_𝐢∩ V_𝐣≠∅ and 2 otherwise, we deduce that χ_𝐢𝐣≥max(χ_𝐢,χ_𝐣)≥ 2.


In order for 𝔼[(a_𝐢-𝔼(a_𝐢))(a_𝐣-𝔼(a_𝐣))] to be non-zero, we must require V_𝐢∩ V_𝐣≠∅ and E_𝐢∩ E_𝐣≠∅. Moreover, each edge in E_𝐢𝐣 must be traversed at least twice to guarantee a nonzero contribution in expectation. This leads to Card(E_𝐢𝐣)≤ k-ℓ and hence 

    Card(V_𝐢𝐣)=1-χ_𝐢𝐣+Card(E_𝐢𝐣)≤ k-ℓ-1.

To sum up, we obtain 

    Var(r_k,1^ℓ,n)≤ 4θ^2 C_k M^2kn^k-ℓ-1n^ℓ=O(4θ^2  M^2kn^k-1)

modulo universal constants depending only on k. This completes the proof.


Now we are in the position to complete the proof of Theorem <ref> in the special case C_n=θ E_11.

 We should first check that the sequence of power series q_n(z) are tight in H(𝔻): this is already verified in <cit.>, Lemma 3.1 when C_n=0, and it is easy to verify that this tightness persists when we add the noise matrix C_n.

Combining all the previous technical lemmas, in particular Lemma <ref>, <ref>, <ref>, <ref>, we conclude that we have the following convergence as holomorphic functions in H(𝔻):

    q_n(z)lawn→∞⟶   (1-zθ)κ(z)e^-F(z)

where κ(z)=√(1-z^2𝔼[a_11^2]) for z∈𝔻 and 

    F(z)=∑_k=1^∞ X_kz^k/k
 where {X_k}_k≥ 1 are independent complex Gaussian variables with 

    𝔼[X_k]=0,  𝔼[|X_k|^2]=1,  𝔼[X_k^2]=𝔼[a_11^2]^k.

We have also used 

    exp(-∑_k=1^∞θ^kz^k/k)=exp(log(1-zθ))=1-zθ.

By definition, κ has no zeros in 𝔻 since |𝔼[a_11^2]|≤ 1.
Since F(z) converges almost surely for any z∈𝔻, we deduce that e^-F(z) has no zeros in 𝔻. That is, provided |θ|>1, the only zero of (1-zθ)κ(z)e^-F(z) in 𝔻 is at z=1/θ.

Since q_n^θ(z) converges almost surely to q_n(z) in ℍ(𝔻), by Rouché's theorem, for n large, almost surely there is only one solution z_n∈𝔻 such that q_n(z_n)=0 and that lim_n→∞z_n=1/θ. In other words, almost surely for n sufficiently large, there is only one eigenvalue of n^-1/2A_n+C_n with modulus larger than 1+o(1), and this eigenvalue converges to θ as n tends to infinity. This proves Theorem <ref> in this special case C_n=θ E_11.




 §.§ The general case of finitely many nonzero elements
 Now we prove the general case of Theorem <ref>, case (1) that covers all C_n which has finite rank and is in a Jordan block form.

 Consider again the trace expansion

    Tr((A_n+√(n)C_n)^k)   =∑_1≤ i_1,i_2,⋯,i_k≤ n,
    |{i_1,⋯,i_k}|=k(a_i_1i_2+√(n)C_i_1i_2)(a_i_2i_3+√(n)C_i_2i_3)⋯ (a_i_ki_1+√(n)C_i_ki_1)
    
       +∑_1≤ i_1,i_2,⋯,i_k≤ n,
     |{i_1,⋯,i_k}|≤ k-1 (a_i_1i_2+√(n)C_i_1i_2)(a_i_2i_3+√(n)C_i_2i_3)⋯ (a_i_ki_1+√(n)C_i_ki_1)

We expand all the brackets on the right hand side, and denote by T_1 the terms consisting of monomials including both terms from a_ij and C_ij and that |{i_1,⋯,i_k}|=k, and let T_2 consist of the monomials including both terms from a_ij and C_ij and that |{i_1,⋯,i_k}|≤ k-1. We wish to prove that T_1/n^k/2 converges to 0 in probability, and that T_2/n^k/2 converges to 0 in probability.

To check the claim concerning T_1, we compute 𝔼[|T_1|^2]. For two typical terms w_𝐢=w_i_1i_2⋯ w_i_ki_1 and r_𝐣=r_j_1j_2⋯ r_j_kj_1 (where each w_ij and r_ij are either some a_ij or some √(n)C_ij), in order for 𝔼[w_𝐢r_𝐣]≠ 0, we require that each edge traversed by some a_ij must be traversed twice (those traversed more than twice has negligible contribution), and this forces there are exactly the same number of pairs in w_𝐢 and r_𝐣 that are some a_ij, and that they must correspond to the same a_ij so as to appear at least twice. In this vein, assume that there are Q edges in w_𝐢 and r_𝐣 traversed by some A_n. We see that to reconstruct this graph we actually need no more than Q-1 free vertices chosen from any of 1,2,⋯,n, as all other vertices are forced to be chosen from the subscript i,j such that C_ij≠ 0, which are finite by definition. Altogether, there are 2(k-Q) other edges traversed by some √(n)C_ij. Thus the overall contribution is O(n^Q-1(√(n))^2(k-Q)), summing over these 𝐢∪𝐣. This readily implies T_1/n^k/2lawn→∞⟶ 0.

To check the claim concerning T_2, in order for a term w_i_1i_2w_i_2i_3⋯ w_i_ki_1 to have non-zero mean, each edge traversed by some a_ij must be traversed twice, so the graph formed by (i_1,⋯,i_k) can be decomposed as (a) a double cycle containing all the terms from A_ij and some other terms from √(n)C_n, and (b) some other cycles around it containing only elements from √(n)C_n. Let R be the number of edges from √(n)C_n, then there are at most k-R/2 edges for the A_ij terms (as each such edge should be traversed at least twice). This corresponds to O(n^k-R/2-1) choices of free vertices from 1,2,⋯,n and a bounded number of choices from vertices i,j such that C_i,j≠ 0. Here we have O(n^k-R/2-1) instead of O(n^k-R/2) choices because at least one vertices from the a_ij's should come from the finitely many subscripts where C_ij≠ 0. The overall contribution to 𝔼[T_2] is thus O(n^k-R/2-1n^R/2) and we have that 𝔼[T_2]/n^k/2 vanishes in the limit. 

Finally we check Var(T_2)=o(n^k) as follows: the computation is exactly the same as before. For two typical terms w_𝐢=w_i_1i_2⋯ w_i_ki_1 and r_𝐣=r_j_1j_2⋯ r_j_kj_1 to have non-zero covariance, each edge traversed by some a_ij should be traversed twice, and assuming there are Q such edges, this leads to Q-1 free vertices and 2(k-Q) other edges traversed by √(n)C_n. This implies Var(T_2)=O(n^k-1). Combining both steps we conclude that T_2/n^k/2lawn→∞⟶0.

To sum up, using also the fact that c_n(z) converges to c(z), and checking the tightness of q_n(z) just as in <cit.>, Lemma 3.1, we have proved that 

    lim_n→∞q_n(z)=κ(z) c(z)e^-F(z)   z∈𝔻,
 in the sense of convergence as functions in H(𝔻), and the proof of Theorem <ref>, case (1) is complete thanks to Rouche's theorem.






 §.§ Infinitesimally shifting the mean
 In this case we consider the matrix C_n such that 

    C_ij=1/nμ
 for each i,j=1,⋯,n.
In this case we have

    Tr((A_n+√(n)C_n)^k)   =∑_1≤ i_1,i_2,⋯, i_k≤ n,
    |{i_1,⋯,i_k}|=k(a_i_1i_2+μ/√(n))(a_i_2i_3+μ/√(n))⋯ (a_i_ki_1+μ/√(n))
       +∑_1≤ i_1,i_2,⋯, i_k≤ n,
     |{i_1,⋯,i_k}|≤ k-1 (a_i_1i_2+μ/√(n))(a_i_2i_3+μ/√(n))⋯ (a_i_ki_1+μ/√(n))
    .

Expanding the product in equation (<ref>), a typical term in the expansion that is not in Tr(A_n)^k or Tr(√(n)C_n)^k is of the form w_𝐢=w_i_1i_2⋯ w_i_ki_1, where r of the w_i_si_s+1 terms are some μ/√(n) and the other k-r terms are some a_ij, for r=1,2,⋯,k-1.


    We have the following convergence in law:

    Tr((A_n+√(n)C_n)^k)-Tr((A_n)^k)-Tr((√(n)C_n)^k)/n^k/2lawn→∞⟶0.



 We first show that the expectation of the left hand side of (<ref>) converges to zero. Recall that a typical term in the numerator of the LHS of (<ref>) is of the form w_𝐢=w_i_1i_2⋯ w_i_ki_1, then in order for 𝔼[w_𝐢]≠ 0, we need that each edge in w_𝐢 that are traversed by some a_ij must be traversed twice (those travelled three times or more are negligible). Thus we can decompose the cycle (i_1,i_2,⋯,i_k,i_1) into (a) a double cycle consisting of all the a_ij's and some μ/√(n)s, and (b) some other cycles attached to this cycle that consist entirely of μ/√(n) terms. Suppose the double cycles in (a) consist of k_1 vertices (and hence k_1 edges), and among the k_1 edges, k_2 of them are traversed by some a_ij's and the other by μ/√(n).  Then there are remaining k-2k_1 edges to be added afterwards. This would account for an addition of strictly less than k-2k_1 new vertices. Thus the overall contribution to the expectation of the left hand side of (<ref>) is, in the case k>2k_1: 

    n^-k/2× O(n^k_1n^k-2k_1-1n^-k-2k_1/2n^-(k_1-k_2))=O(n^-1),

and in the case k=2k_1, one must have k_1>k_2, so that the contribution is

    n^-k/2× O(n^k_1n^-(k_1-k_2))=O(n^-1).

This proves the expectation of the left hand side of (<ref>) converges to 0.

We then show the variance of the left hand side of (<ref>) converges to 0 as well. For two typical terms w_𝐢=w_i_1i_2⋯ w_i_ki_1 and r_𝐣=r_j_1j_2⋯ r_j_kj_1 to have non-zero covariance, we consider the graph formed by 𝐢∪𝐣, whose vertices and edges are the union of those from 𝐢 and 𝐣. Then each edge traversed by some a_ij in 𝐢∪𝐣 should be traversed twice (those travelled more times are negligible), and such that there are equal number of terms in w_𝐢,r_𝐣 travelled by a_ij's, and they have the same subscript. Therefore 𝐢∪𝐣 is made up of (a) a double cycle with k_1 edges (each travelled twice) and k_1 vertices, of which k_2 are travelled by the a_ij's; and (b) 2k-2k_1 outlying edges, giving rise to at most 2k-2k_1-1 new vertices. Then the overall contribution to the variance of the LHS of (<ref>) is at most 

    n^-k× O(n^k_1n^2k-2k_1-1n^-2k-2k_1/2n^-(k_1-k_2))=O(n^-1),
 in the case when k<k_1,
and 

    n^-k× O(n^kn^-(k_1-k_2))=O(n^-1),

in the case when k=k_1 so that k_1>k_2,
completing the proof.




Thanks to Lemma <ref>, and proving tightness of q_n(z) just as the previous cases, we conclude that 

    q_n(z)lawn→∞⟶(1-μ z)κ(z)e^-F(z),   z∈𝔻,

regarded as convergence of holomorphic functions in H(𝔻), so that when |μ|>1, almost surely as n gets large, there is precisely one eigenvalue of n^-1/2A_n+C_n with modulus larger than 1+o(1), and this eigenvalue converges to μ as n tends to infinity. 

More generally, if we assume that |C_i,j|≤M/n for some fixed M>0 and each i,j=1,⋯,n, then most previous arguments still apply, and Lemma <ref> is still valid. Thanks to the assumption that c_n(z), the reverse characteristic function of C_n, converges to c(z) in H(𝔻), we deduce that

    q_n(z)lawn→∞⟶c(z)κ(z)e^-F(z),   z∈𝔻,

so that by Rouche's theorem, eigenvalues with modulus larger than 1+o(1) converge to those z∈ℂ that solves c(z^-1)=0 and |z|>1, and vice versa.
This proves Theorem <ref> under the case (2) stated there.




§ PERTURBATION OF SPARSE I.I.D. MATRICES

Now we are in the setting to prove Theorem <ref>. We recall the trace asymptotic of the random matrix A_n from <cit.>: 



 
 The random matrix A_n in the definition of Theorem <ref> satisfies the following trace asymptotic:

  * In case (1) of Theorem <ref>(sparse with mean degree d), let Y_ℓ,ℓ∈ℕ^* a family of independent Poisson random variables with Y_ℓ∼Poi(d^ℓ/ℓ), and define 

    X_k:=∑_ℓ| kℓ Y_ℓ,   (k∈ℕ^*).
 Then we have the joint weak convergence 

    (Tr(A_n^1),⋯.Tr(A_n^k))lawn→∞⟶      (X_1,⋯,X_k).



  * In case (2) of Theorem <ref>(semi-sparse with mean degree d_n→∞,d_n=n^o(1)), let (N_k:k≥ 1) be a family of independent real Gaussian random variables with mean 0, variance 1. Then for any k, we have the joint convergence in law

    (Tr(A_n)/√(d_n)-√(d_n),⋯, Tr(A_n^k)/√(d_n)^k-√(d_n)^k  )
        lawn→∞⟶      (N_1,⋯,√(k)N_k)+(0,1,0,1,⋯ ,1_k even ).


    


In this sparsity regime we prove the following simple yet crucial lemma:

 Assume the deterministic matrix C_n satisfies the assumptions made in Theorem <ref>. Then,
    
        
  * In case (1) of Theorem <ref>, for each k we have the following convergence in law
        
    Tr((A_n+C_n)^k)-Tr(A_n^k)-Tr(C_n^k)    lawn→∞⟶ 0.

        
  * In case (2) of Theorem <ref>, for each k we have the following convergence in law
        
    Tr((A_n+√(d_n)C_n)^k)-Tr(A_n^k)-Tr((√(d_n)C_n)^k)    lawn→∞⟶ 0.

    



    In both cases, expanding the trace of matrix power, the terms contributing to the left hand side of (<ref>), (<ref>) are of the form
    
    ∑_1≤ i_1,⋯,i_k≤ n, w_i_1i_2w_i_2i_3⋯ w_i_ki_1,

where for some s, we have w_i_si_s+1=a_i_si_s+1 and for some s' we have w_i_s'i_s'+1=(√(d_n))C_i_s',i_s'+1 (for case (1), we don't need the √(d_n) term, but this is irrelevant to the forthcoming proofs). That is, both terms from A_n and C_n should be present in this product. The sequence (i_1,i_2,⋯,i_k,i_1) must form a cycle, and hence is a union of edge disjoint directed loops (including self loops). Suppose t distinct indices are involved in the formation of this cycle, then there must be at least t distinct edges involved in the cycle. Suppose that m of these distinct edges are traversed by an element (d_n)^1/2C_ij, then since both vertices of such edges should come from indices where C_ij≠ 0 and there are only finitely many such vertices, we see that either (a) at least m+1 vertices are determined by these finitely many choices, or (b) these  edges traversed by √(d_n)C_j,k should form a cycle. In case (a), we have only O(n^t-m-1) choices of free vertices. Meanwhile, for each i,j we have 𝔼[a_ij]=d_n/n, so the overall expectation of (<ref>) is 
    O(n^t-m-1(d_n/n)^t-m(d_n)^m)=O((d_n)^2k/n).
 Since by assumption d_n=n^o(1), we see that contribution of such terms to the expectation of (<ref>) converges to zero. In case (b),we need to add additional cycles on the cycle already formed by the edges from √(d_n)
C_ij: we first add in those cycles involving edges from √(d_n)C_ij, and this case is exactly the same as in case (a), and what remains is cycles involving purely the a_ij terms.   But we can see that an addition of s vertices for such remaining cycles lead to at least s+1 more edges traversed by a_ij, and such terms contribute O((d_n)^2k/n) to the expectation of (<ref>),(<ref>) and is negligible as well. Combining both cases completes the proof.




Now we can complete the proof of Theorem <ref>.


We begin with case (1) of Theorem <ref>.
    Denote by 
    q_n(z)=(I-z(A_n+C_n)).
 Then modifying the arguments in <cit.>, one can show that {q_n(z)}_n≥ 1 are tight as holomorphic functions on {z∈ℂ:|z|≤1/√(d)}. By <cit.>, Theorem 2.7 with a slight modification, combining the computations in Lemma <ref>, we deduce that
    
    q_n(z)lawn→∞⟶e^-f(z),

    where 
    
    f(z)=∑_k=1^∞ (X_k+Tr(A_n^k))z^k/k,

    and this expression can be simplified as follows: denoting by
    
    ℛ(z)=∑_k=1^∞ (τ_k-X_k)z^k/k,  τ_k=𝔼[X_k],

    we have 
    
    q_n(z)lawn→∞⟶e^ℛ(z)× c(z)×∏_ℓ=1^∞ (1-dz^ℓ)^1/ℓ,

    and the convergence is in terms of convergence of holomorphic functions on {z∈ℂ:|z|<d^-1/2}. By <cit.>, section 8, the factor e^ℛ(z) is nonzero everywhere on {z∈ℂ:|z|<d^-1/2}, so by Rouche's theorem, almost surely for n sufficiently large, the only eigenvalues of A_n+C_n with modulus larger than √(d)(1+o(1)) converge either to d or to those z∈ℂ satisfying c(z^-1)=0 and |z|>√(d), and each root of the latter equation corresponds to the n→∞ limit of some outlier eigenvalues λ_n of  A_n+C_n.

    Now we prove case (2) of Theorem <ref>. For this it suffices to consider 
    
    q_n(z)=(I_n-z(d_n^-1/2A_n+C_n)),

    Define 
    
    G(z)=exp{∑_k=1^∞ N_kz^k/√(k)},
 then by modifying the proof of <cit.>, Lemma 10.1, we can show q_n(z)/√(d_n) is tight in H(𝔻). Then modifying the proof of <cit.>, Theorem 3.2, we have the convergence
    
    q_n(z)/√(d_n)lawn→∞⟶ -zc(z)√(1-z^2)G(z).

    in terms of convergence as functions on H(𝔻). Therefore by Rouche's theorem, almost surely for n sufficiently large, the eigenvalues of (d_n)^-1/2A_n+C_n having modulus larger than 1+o(1) either converge to infinity (with multiplicity one), or converge to roots of c(z^-1)=0 that satisfies |z|>1. Conversely, all the roots to c(z^-1)=0 with |z|>1 arise as the n→∞ limit of some outlier eigenvalues λ_n of (d_n)^-1/2A_n+C_n. This completes the proof.










 Paper Title: Remarks on the Poisson additive process.





Remarks on the Poisson additive process]Remarks on the Poisson additive process






Wang Haoming]Wang Haoming
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
wanghm37@mail2.sysu.edu.cn










[2020]Primary 60Axx, 60Gxx; Secondary 60A05, 60G05, 60G55.




§ ABSTRACT 
	The Poisson additive process is a binary conditionally additive process such that the first is the Poisson process provided the second is given. We prove the existence and uniqueness of predictable increasing mean intensity for the Poisson additive process. Besides, we establish a likelihood ratio formula for the Poisson additive process. It directly implies there doesn't exist an anticipative Poisson additive process which is absolutely continuous with respect to the standard Poisson process, which confirms a conjecture proposed by P. Brémaud in his PhD thesis in 1972. When applied to the Hawkes process, it concludes that the self-exciting function is constant. Similar results are also obtained for the Wiener additive process and Markov additive process.


[
    [
    
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§ INTRODUCTION

	
	In 1937, Paul Lévy<cit.> introduced the concept of additive processes which are stochastically continuous and have the property of independent increments. In this note, we generalize the additive process to the binary conditionally additive process which is characterized as 
	
		
  * {(X_t,Y_t)}_t ≥ 0 is continuous in probability, that is, for any t ≥ 0 and ε >0
		
    lim_s→ t𝐏(|X_s - X_t|+ |Y_s-Y_t|≥ε)=0,

		
  * X_t has conditionally independent increments given Y_t, t≥0, that is, for any 0 ≤ t_0 < t_1 < t_2 < … < t_n the random variables X_t_1-X_t_0, X_t_2-X_t_1, ⋯, X_t_n-X_t_n-1 are conditionally independent provided that the σ-algebra generated by the family of random variables Y_t, t≥ 0 is given.
	
	
	Recall two random variables X and Y are conditionally independent under the σ-algebra 𝒢⊆ℱ on a probability space (Ω,ℱ,𝐏) if 
	
    𝐄[XY|𝒢] = 𝐄[X|𝒢]𝐄[Y|𝒢].

	The case for n variables is also similar. Let {ℱ_t}_t≥ 0 be an increasing family of σ-algebras on the probability space (Ω,ℱ,𝐏) satisfying ℱ_t⊆ℱ for each t. The conditionally additive process (X_t,Y_t) is adapted to ℱ_t if X_t is measurable with respect to ℱ_t for each t≥ 0 and
	
    σ(Y_t:t≥ 0) ⊆ℱ_0,

	i.e. the σ-algebra generated by Y_t, t≥ 0 is included in ℱ_0. We will assume that {(X_t,Y_t)}_t≥ 0 is adapted to filtration {ℱ_t}_t≥ 0 in this article.
	
	Our question now becomes two-fold, that is, under what conditions
	
		
  (1) there exists 𝒢_t⊇ℱ_t for each t≥ 0 such that ℱ_t can be expanded to a larger one 𝒢_t preserving  martingale, conditionally independent increment and Markov properties etc.
		
  (2) there exists 𝒢_t⊆ℱ_t for each t≥ 0 such that ℱ_t can be restricted to a smaller one 𝒢_t preserving martingale, conditionally independent increment and Markov properties etc.
	
	
	The first question has been answered by many authors, such as K. itô, M. T. Barlow, T. Jeulin, M. Yor and P. Protter who are the one wrote a chapter on this topic, c.f. <cit.>, while the second according to the author still remains unknown. This article attempts to solve the latter problem in a quite general setting, especially the Poisson restriction problem proposed by P. Brémaud<cit.> who asks when filtration is the natural one, i.e. the one generated by the internal process itself. 
	
	The adapted process is anticipative if there exists t≥ 0 such that 
	
    ℱ_t≠σ (X_s: 0 ≤ s ≤ t).

	Otherwise, it is non-anticipative if filtration coincides with the intrinsic σ-algebra generated by this process, i.e. for any t ≥ 0,
	
    ℱ_t = σ (X_s: 0 ≤ s ≤ t)

	A closely related concept for the adapted process is self-exciting,  if for any t ≥ 0,
	
    σ (X_s: 0 ≤ s ≤ t) ⊆ℱ_t⊆σ (X_t: t ≥ 0)
 
	is fulfilled. P. Brémaud in 1972 conjectured in his PhD thesis<cit.> that there dosen't exist an anticipative self-exciting process which is absolutely continuous with respect to the standard Poisson process. We shall give an affirmative answer to this. In fact, we will show more generally comparison with respect to the standard model yields a non-anticipative filtration. Here the standard model stands for the Brownian motion, standard Poisson process, and stationary Markov chain. The measure induced from an adapted process governed by the standard one hence solves the problem (2) of restriction of filtration. These processes are our main concern in this article with the likelihood ratio given by the Radon-Nikodym derivative. 
	
	In conclusion, the adaptiveness happens if we could predict the future based on extra information plus memory about the present and before. For a binary conditionally additive process, the non-anticipativeness usually leads to many useful consequences since in this case the second ancillary process shall degenerate to a non-stochastic function, without possessing non-deterministiness anymore. For the Hawkes process, the self-exciting term then becomes constant. In this sense we will focus on the contrary that any prediction about future should only depend on the memory until this present. This is crucial in biological study, astronomical observation, economic calculation and many other applications in our daily life. 
	
	
	
	
	
	
	
	
	
	

§ THE POISSON ADDITIVE PROCESS
 
	
	The anticipative Poisson additive process, which is also the self-exciting process plays a central role in various statistical applications. For example, the Cox process, Neymann-Scott process and Hawkes process etc. According to Snyder<cit.>: "the self-exciting process is a binary conditionally additive process with non-independent increments such that the number of arrival times and arrival times after time t depends on the number of arrival times N_t before time t and the arrival times 0 = t_0 < t_1 < t_2 < … < t_N_t." It usually appear in engineering literature, like operations research, queuing and dis-patching problems. 
	
	
	More formally, the Poisson additive process is characterized as a binary conditionally additive process {(N_t,A_t)}_t ≥ 0 satisfying the following properties
	
		
  2.A (N_t, A_t) is adapted to ℱ_t.
		
  2.B the random variable N_t - N_s has a conditional Poisson distribution for each 0 ≤ s < t,
		
    𝐏(N_t - N_s = n|ℱ_s) = (A_t - A_s)^ne^A_s - A_t/n!,    n =0, 1,2,…

		with 0^0 =1 in mind. 
		
  2.C the random variables N_t - N_s and N_u - N_v are conditionally independent given ℱ_s for 0 ≤ s < t ≤ u < v.
	
	
	In fact, the definition 2.B already implies (<ref>), a condition crucial in subsequent analysis that hasn't been noticed in recent articles. This definition 2.A-C means that N_t is an adapted point process with conditionally dependent increments such that N_t-N_s has the Poisson distribution with mean A_t - A_s given the mean intensity A_t, t≥0. 
	
	[Hawkes process] 
		Suppose N_t have arrivals at times 0=t_0<t_1<t_2<t_3<⋯ where the infinitesimal probability of an arrival during the time interval (t,t+dt] is 
		
    λ _t dt=(μ (t)+∑ _t_i:t_i≤ tϕ (t-t_i))dt.
 
		where μ is the underlying baseline intensity and ϕ is the self-exciting term. It is a Poisson additive process if 
		
    σ(t_n: n ≥ 0) ⊆ℱ_0
 
		but it may not be non-anticipative.
	
	
	In 1972, P. Brémaud<cit.> has raised a question whether there exists an anticipative self-exciting process which is absolutely continuous with respect to the standard Poisson process. We shall give a negative answer to his question. To achieve this, we need a concept of predictable increasing process A_t such that the difference with N_t and A_t is a martingale, i.e. 
	
    𝐄[N_t - A_t|ℱ_s] = 𝐄[N_s - A_s|ℱ_s],    for each  0 ≤ s ≤ t,

	If either N_t or A_t the expectation of which is finite for each t ≥ 0, (<ref>) becomes
	
    𝐄[N_t - N_s|ℱ_s] = 𝐄[A_t - A_s|ℱ_s],    for each  0 ≤ s ≤ t.

	Intuitively, the adapted point process N_t and the predicatble increasing mean intensity A_t share the same mean. It is known that N_t - A_t is a martingale equivalent to N_t a Poisson process with mean intensity A_t. In 1964, S. Watanabe proved the case when A_t is constant and in 1975, P. Brémaud proved the case when A_t is absolutely continuous with respect to the Lebesgue measure<cit.>. However, problem comes if we pay attention to this more concrete example.
	
		Let a > 0 and  
		
    λ_t =  1 ,    if  N_t+a - N_t > 0,
    
    				2 ,    if  N_t+a - N_t = 0.

		Then
		
    ℱ_t = σ{N_s: 0 ≤ s ≤ t}σ{N_t+a - N_t>0}⊆σ{N_s:s < t+a},

		while N_t -t is not a 𝒢_t = σ{N_s: 0 ≤ s ≤ t}σ{N_t+a - N_t=1}-martingale. 
	
	The properties of conditionally independent increments of the Poisson additive process hence highly relies on the choice of filtration. Obviously the process given by (<ref>) is Poisson additive with respect to ℱ_t given by (<ref>) but isn't to 𝒢_t. The following theorem generalizes the martingale characterization of S. Watanabe and P. Brémaud<cit.>.
	
	 Let N_t be an adapted point process with predictable increasing mean intensity A_t such that 
		
    EA_T < ∞ for fixed T>0

		and (<ref>) holds true on [0,T]. (N_t,A_t) is a Poisson additive process with predictable increasing mean intensity A_t if and only if N_t - A_t is a martingale.
		
		
		
		
		
			
				
				
			
			
		
		
		
		
		
	 
	
	 
		
			
			
			
			
		
		
		
		Necessity is guaranteed by the definition 2.B. To prove sufficiency, first note that A_t	is bounded on [0,T] almost surely. By applying a version of the itô formula for the jump process,
		
    exp(iuN_t) = exp(iuN_0) + ∑_0 < s ≤ t (e^iu - 1) exp(iuN_s-) Δ N_s,
	and substituting M_t = N_t - A_t in (<ref>), we obtain
		
    exp(iuN_t) - ∫_0^t (e^iu - 1) exp(iuN_s-) dA_s
is a martingale since the integration about predictable processes preserves the martingale property. Therefore, taking the difference between the upper limit of the integral in (<ref>) at t and s, we get
		
    exp(iuN_t) - exp(iuN_s) - (e^iu - 1) exp(iuN_s) ∫_s^t exp(iu(N_v- - N_s)) d A_v.

		Dividing both sides of (<ref>) by exp(iuN_s) and taking the conditional expectation, we have
		
    𝐄[exp(iu(N_t - N_s)) | ℱ_s] = 1 + (e^iu - 1) 	𝐄[∫_s^t exp(iu(N_v- - N_s)) dA_v | ℱ_s].
	
		By interchanging the order of integration in (<ref>) and noting condition (<ref>), we get
		
    𝐄[exp(iu(N_t - N_s)) | ℱ_s] = 1 + (e^iu - 1) ∫_s^t 𝐄[exp(iu(N_v- - N_s)) | ℱ_s] dA_v.

		Denoting the left-hand side of (<ref>) as y_t, (<ref>) becomes
		
    y_t = 1 + (e^iu - 1) ∫_s^t y_v- dA_v.
	Taking the derivative of both sides of (<ref>) with respect to t and noting the properties of Lebesgue integration, y_t is the solution of the following ordinary differential equation:
		
    dy_t/dt = (e^iu - 1) A^'_t y_t,     t > s, 
    
    				y_s = 1,     t = s.
	
		The differential equation (<ref>) has the explicit solution
		
    y_t = e^(e^iu - 1) (A_t - A_s) ,    t ≥ s.

		This proves that N_t is a Poisson additive process with mean intensity A_t.
	
	
	 The theorem still holds if the condition 4 is weakened to local martingales c.f. <cit.>.
	
	
	
		The uniqueness of predictable increasing mean intensity A_t then follows from the Doob-Meyer decomposition theorem.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	[One-shot point process]  Suppose X has an exponential distribution, say F(x)= 𝐏 (X ≤ x) = 1 - e^-λ x. Then the corresponding one-shot point process has mean intensity A_t=λmin(t,X) which is differentiable except at X, and in any case is absolutely continuous with density 
    A_t^' = λ   , t ≤  X
    
    				0    , t>X
	
		This can be evaluated via the IHF in the following section.  
		
	
	
	
	

§ INTEGRATED HAZARD FUNCTION

	
	There are several ways to construct the predictable increasing mean intensity of an adapted point process of which the uniqueness is guaranteed by the Doob-Meyer decomposition theorem. Next we illustrate a result using the so-called integrated hazard function (IHF) concerning the existence of such processes. This assumes the conditional distribution of waiting times, i.e. the difference of arrival times exists and has various useful applications. Let T_n be the arrival times for the point process N_t, i.e.
	
    T_n= inf{t ≥ 0: N_t≥ n},

	or equivalently, 
	
    T_n= sup{t ≥ 0: N_t <n}.

	For a stopping time T, defnite the σ-algebra
	
    ℱ_T= {A ∈ℱ: A ∩{T ≤  t}∈ℱ_t for all  t ≥ 0}.

	 
		Suppose the regular conditional distribution F_n(ω,du) of waiting times U_n+1 = T_n+1 - T_n given ℱ_T_n exists. Then a predictable increasing version of the mean intensity is 
		
    A_t = ∑_n=0^∞ A^(n)_t,
 
		where A_t^(n) is given by the usual hazard function increasing on the real line
		
    A^(n)_t = ∫_0^t - T_nF_n (dx)/1 - F_n(x-) , T_n < t ≤ T_n+1,    const.  otherwise.
 
	
	 Suppose N_t is a point process and A_t a predictable increasing process. For any non-negative predictable process C_t on [0,T],
		
    𝐄[∫_0^TC_td N_t] = 𝐄[∫_0^TC_td A_t]

		if and only if	 N_t - A_t is martingale on [0,T].
	
	 Choose a predictable generator C(u,ω) = 1(s < u ≤ t)1(ω∈Λ), Λ∈ℱ_s and this follows from monotone class theorem.
	
	
		By Lemma 3.2, it suffices to show that for each n and finite stopping time  T, E[N_T T_n] = E[A_T T_n], with A given by (<ref>).
		
			Let T be a σ(N_s:0≤ s ≤ t)-stopping time. For each n there is a random variable R_n∈σ(N_T_n s:s≥ 0) such that on {T_n < T},
			
    T  T_n+1 = (T_n + R_n )  T_n+1

		
		 It is direct to verify these two equalities
			
    σ(N_T_n s:s≥ 0)   = σ(T_1,⋯,T_n)
    σ(N_T s:s ≥ 0)∩{T_n≤ T < T_n+1}   = σ(N_T_n s:s ≥ 0)∩{T_n≤ T < T_n+1}

			In fact we can take R_n = [g_n(T_1,⋯,T_n) -T_n]_+ where g_n=T on {T_n < T < T_n+1}.
		
		
		Return to our proof now. On the one hand, Claim 3.3 allows us to truncate the series (<ref>) into finite sums,
		
    𝐄[A_T T_n]     = 𝐄[∑_j=0^n-1[∫_0^R_j U_j+1dF_j(x)/1-F_j(x-)]1(T ≥ T_j)]
        = ∑_j=0^n-1𝐄[ 𝐄[.∫_0^R_j U_j+1dF_j(x)/1-F_j(x-)|ℱ_T_j]1(T ≥ T_j)]
        = ∑_j=0^n-1𝐄[∫_0^R_jF_j(dx)1(T ≥ T_j)]

		since R_n∈σ(N_T_n s:s≥ 0). But on the other hand, on {T ≥ T_j},
		
    N_T T_j+1 - N_T T_j = 1(R_j≥ U_j+1)1(T ≥ T_j) = ∫_0^R_jF_j(dx)1(T ≥ T_j)

		Take expectation and sum these terms in (<ref>) 
		
    𝐄[N_T T_n] = ∑_j=0^n-1𝐄[∫_0^R_jF_j(dx)1(T ≥ T_j)]

		too. Compare (<ref>) and (<ref>) to derive the desired result.
	
	
	
		Let N_t be an adapted point process with predictable increasing mean intensity A_t. 
		Then for each t,
		
    A_t= lim_n →∞∑_k=0^2^n-1𝐄[N_(k + 1)t/2^n - N_kt/2^n|ℱ_kt/2^n]

		holds almost surely.
	
	
	
		Let T > 0 be fixed and 𝐏, 𝐏^' be two probability measures on (Ω, ℱ_T) with respect to which the point process has common mean intensities. Then 𝐏 = 𝐏^' on ℱ_T provided that 𝐏 = 𝐏^' on ℱ_0.	
	
	
	
	
	 <cit.>
		
	
	
	
	
	
	

§ THE RADON-NIKODYM DERIVATIVE
 
	
	The instantaneous intensity is a predictable non-negative process λ_t satisfying A_t = ∫_0^tλ_sds. We already proved the uniqueness and existence of predictable increasing mean intensity of an adapted point process. In this section we are going to prove there exists an intrinsic instantaneous intensity for the Poisson additive process under suitable conditions, for instance, provided the likelihood ratio formula given by the Radon-Nikodym derivative.
	
	
		Let N_t be a Poisson additive process with σ(N_s:0≤ s ≤ t)- instantaneous intensity λ_t. Suppose that 𝐏_0 is a probability measure on (Ω, ℱ_T) with respect to which N_t is a stationary Poisson process with rate 1. Then 𝐏≪𝐏_0 and for each t ∈ [0,T],
		
    .d𝐏/d𝐏_0|_σ(N_s:0≤ s ≤ t) = exp[∫_0^t (1 - λ_s )ds + ∫_0^tlogλ_s d N_s]

		Conversely, if 𝐏_0 is as above and 𝐏 is a  probability measure on (Ω, ℱ_T)  satisfying 𝐏≪𝐏_0, then there exists a σ(N_s: 0 ≤ s ≤ t)-predictable process λ_t such that N_t has instantaneous intensity λ_t with respect to 𝐏.
	
	
	 
		Sufficiency. For u < t define 
		
    Z_u = exp[∫_0^u (1-λ_s)ds + ∫_0^ulogλ_s d N_s]

		By property of independent increments of Poisson processes that (Z_u)_u ≤ t is a martingale over (Ω, ℱ_T, 𝐏_0). Then by lemma 3.2, it suffices to show that if  𝐏^' is the probability measure defined by d𝐏^' = Z_td𝐏_0, then for each bounded, predictable process C,
		
    𝐄^'[∫_0^t C_s dN_s] = 𝐄^'[∫_0^t C_sλ_sds]
 
		Since Z_t is a 𝐏_0-martingale, (<ref>) reduces to 
		
    𝐄_0[∫_0^tZ_sC_sdN_s] = 𝐄_0[∫_0^tZ_sC_sλ_sds]

		Plug the relation Z_T_n = Z_T_n-λ_T_n for arrival time T_n in the LHS of (<ref>) and note that almost surely Z_s- = Z_s for every s∈ [0,t] except an at most countable set which is Lebesgue null. Then this relation holds true since under 𝐏_0 the rate of N is 1.

		
		Necessity. We need to show that if 𝐏≪𝐏_0 then there is a σ(N_s:0≤ s ≤ t)-predictable process λ_t that is the instantaneous intensity of N_t with respect to 𝐏. Define the 𝐏_0-martingale Z_t = 𝐄_0[.d𝐏/d𝐏_0|σ(N_s: 0 ≤ s ≤ t)]. 
		According to a martingale representation theorem in the sense of J. Jacod<cit.>, there is a σ(N_s:0≤ s ≤ t)-predictable process H_t such that
		
    Z_t = 1 + ∫_0^t H_s(dN_s -ds).

		Given a bounded, predictable process C_s vanishing  outside [0,t], 
		
    𝐄[∫_0^t C_s dN_s] = 𝐄_0[Z_t∫_0^t C_s dN_s]

		Since Z_t is a 𝐏_0-martingale, the RHS of (<ref>) equals
		
    𝐄_0[∫_0^t(Z_s- + H_s)C_s dN_s]= 𝐄[∫_0^t (1+ H_s/Z_s-)1(Z_s->0) C_s ds]

		Here we use the fact that Z_T_n =  Z_T_n- + H_T_n at each arrival time T_n. Since the positive martingale Z_t is trapped at zero if it ever reaches zero, in which case, by (<ref>), H_t becomes zero as well. Take λ_t = (1 + H_t/Z_t-)1(Z_t- > 0). 
	
	
	 If there exists a probability measure 𝐏_0 on (Ω,ℱ_T) with respect to which a Poisson additive process N_t has the standard Poisson and 𝐏≪𝐏_0, then it is non-anticipative in the sense of (<ref>) fulfilled.  In particular, there doesn't exist an anticipative self-exciting process which is absolutely continuous with respect to the standard Poisson process in the sense of (<ref>) fulfilled.
	
	
	[Continued to Example 2.1] 
		Suppose N_t is a Poisson additive process in Example 2.1 satisfying the condition of Corollary 4.2
		
    λ _tdt=(μ (t)+∑ _t_i:t_i≤ tϕ (t-t_i))dt.
 
		where μ is the underlying intensity and ϕ is the self-exciting function. The first arrival occurs at time t_1 and immediately after that, the intensity becomes 
		μ (t)+ϕ(t) + ϕ (t-t_1), and at the time t_2 of the second arrival the intensity jumps to μ (t)+ϕ(t) + ϕ (t-t_1)+ϕ (t-t_2) and so on. By corollary 4.2, ϕ is constant when the above assumptions are satisfied.
	
	
	It deserves to note that the likelihood ratio method indeed has a very long history. In fact, in two papers on Proceedings of the National Academy of Sciences published by C. Kraft in 1955 and L. Shepp in 1964 independently we know that any two Gaussian process is either mutually equivalent or perpendicular.  With the aid of dichotomy theorem for Gaussian measures, there are a bundle of necessary and sufficient conditions for Gaussian process equivalent to another Gaussian process, especially the Wiener process. One of them involving the likelihood ratio will be introduced in the next section.
	
	Moreover, we've seen the key point to calculate the likelihood ratio is the martingale predictable representation. The classical martingale representation theorem states that for any local martingale adapted to non-anticipative filtration generated by the Brownian motion, there exists a predictable process that is the integrand of this process to the Brownian motion. The proof just inherits from that of Brownian motion for a Markov chain on the finite state space, for example, under suitably nice conditions the integration can be performed in terms of a solution of the martingale problem. However, distinct from the martingale representation theorem above, that of Poisson process, which relies on the Doob's optional sampling theorem, is much harder. The readers can find a proof in <cit.>.
	
	
	
	

§ APPLICATIONS TO THE WIENER ADDITIVE PROCESS AND THE MARKOV ADDITIVE PROCESS

	
	
	In this section we generalize the non-anticipative result for Poisson additive process to the Wiener additive process and Markov additive process, and about both of which conditional independence or Markov additive property is understood in conditioning on an increasing family of σ-algebras {ℱ_t}_t ≥ 0 satisfying ℱ_t⊆ℱ for each t ≥ 0. More precisely, the Wiener additive process is characterized as a continuous process W_t with covariance R_t,s satisfying
	
		
  5.A W_t is adapted to ℱ_t and 
		
    σ(R_t,s: t,s ≥ 0) ∈ℱ_0.

		
  5.B the random variable W_t - W_s has a conditional normal distribution for each 0 ≤ s ≤ t, a < b
		
    𝐏 ( a< W_t - W_s≤ b |ℱ_s) = ∫_a^b1/√(2πσ^2)exp{-y^2/2σ^2}dy
  
		where σ^2 = R_t,t + R_s,s - R_s,t - R_t,s.
		
  5.C if s=t_0 < t_1 < t_2 < … < t_n the random variables W_t_1-W_t_0, W_t_2-W_t_1, ⋯, W_t_n-W_t_n-1 are conditionally independent given ℱ_s.
	
	where, besides, the R_t,s is 
	
		
  * symmetric R_t,s = R_s,t, 
		
  * locally square-integrable  
		∫_0^T∫_0^TR_s,t dt ds < ∞   a.s.  for fixed  T > 0,
		
  * and positively definite 
		∑_i,j=1^n c_ic_jR_t_i,s_j≥ 0,
		for all real numbers c_i,c_j and t_i,t_j≥0.
	 
	
	In a word, the Wiener additive process describes a perturbed system that needs extra information to predict the future at the present or W_t is an adapted continuous process with conditionally dependent increments such that W_t-W_s has the normal distribution with variance σ^2 given the conditional covariance R_t,s, t,s ≥ 0. Another equivalent formalization is
	
		
  5.A^' (W_t,A_t) is a binary conditionally additive process adapted to ℱ_t.
		
  5.B^' the random variable W_t - W_s has a conditional normal distribution for each 0 ≤ s ≤ t, a < b
		
    𝐏 ( a< W_t - W_s≤ b |ℱ_s) = ∫_a^b1/√(2πσ^2)exp{-y^2/2σ^2}dy
  
		where σ^2 = A_t - A_s.
		
  5.C^' if s=t_0 < t_1 < t_2 < … < t_n the random variables W_t_1-W_t_0, W_t_2-W_t_1, ⋯, W_t_n-W_t_n-1 are conditionally independent given ℱ_s.
	
	where the conditional variance A_t is a positive, locally square-integrable process.
	
	It's easily calculated from definition 5.C^' that R_t,s = A_min(t,s) so these two definitions truly coincide. For convenience, we will adopt the latter definition in the sequel. If the process starts at zero W_0 = 0 with A_0 = 0 and the variance A_t is increasing almost surely, we can derive from 5.B the standard Brownian motion by multiplying a scaling coefficient B_t = (A^'_t)^-1/2 W_t so there always exists a probability measure 𝐏_0 on (Ω,ℱ_T) with respect to which the Wiener additive process has the standard Brownian motion, say
	
    𝐏(x_1, …, x_n) = 𝐏(ω: W_t_1(ω) ≤ x_1,…,W_t_n(ω) ≤ x_n),

	and 	
	
    𝐏_0(x_1, …, x_n) = 𝐏(ω: W_t_1(ω)/√(A^'_t_1(ω) )≤ x_1,…, W_t_n(ω)/√(A^'_t_n (ω))≤ x_n).

	By the dichotomy of Gaussian measures we have the following characterization.
	 Let W_t be a Wiener additive process with conditional variance A_t. In order that 𝐏 is equivalent to 𝐏_0, for brevity we write 𝐏 || 𝐏_0, A^'_t must equal to 1 except a Lebesgue measure zero set.
	
	
	
	
		<cit.>.
	
	
	Second Proof. We shall give another proof for completeness. Suppose ϕ_t is 
	
		
  * progressively measurable ℬ([0,t]) ×ℱ_t measurable for each  t ≥ 0,
		
  * locally square-integrable ∫^T_0ϕ^2_tdt < ∞ a.s. for fixed  T > 0.
	 
	The velocity is a progressive locally square-integrable process ϕ_t such that 
	
    W_t = B_t - ∫_0^tϕ_sds.

	where B_t is the standard Brownian motion.
	
	
	
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
	
	
	
	
	
	
	
		Let (W_t,A_t) be a Wiener additive process with σ(W_s:0≤ s ≤ t)-velocity ϕ_t and B_t the standard Brownian motion. Suppose 𝐏_0 is a probability measure on (Ω, ℱ_T) with respect to which W_t is the standard Brownian motion, then 𝐏 || 𝐏_0 and there exists a σ(W_s:0≤ s ≤ t)-stopping time τ such that 
		
    .d 𝐏/d 𝐏_0|_σ(W_s:0≤ s≤ t) = exp[∫_0^tϕ_s dB_s - 1/2∫_0^tϕ_s^2ds],    t < τ

		Conversely, if there exists 𝐏_0 as described above and 𝐏|| 𝐏_0, then there exist a σ(W_s: 0 ≤ s ≤ t)-velocity ϕ_t on [0,T] and a σ(W_s: 0 ≤ s ≤ t)-stopping time τ such that (<ref>) holds. 
	
	 To begin with, let Z_t = 𝐄_0[.d 𝐏/d 𝐏_0|σ(W_s:0≤ s≤ t)]. If 𝐏 and 𝐏_0 are equivalent, it's not difficult to verify that Z_t > 0 a.e. 𝐏 and 𝐏_0. Define a stopping time τ = inf{ t ≤ T : Z_t = 0 }. By noting that Z_t is a positive supermartingale, we have Z_t = 0 for t ≥τ. We shall prove Z_t has the form
		
    Z_t = exp{∫_0^t ϕ_s dB_s - 1/2∫_0^t ϕ_s^2 ds },    t < τ.

		It is shown that Z_t is a local martingale of Brownian motion. Then by the classical martingale representation theorem in the sense of Kunita-Watanabe<cit.>, Z_t can be written as a stochastic integral,
		
    Z_t = 1 + ∫_0^t g_s dB_s,

		where g_t is a σ(W_s:0≤ s≤ t)-progressively measurable and square-integrable process. However, we can make the same assertion for g_t a.s. 𝐏 since 𝐏 || 𝐏_0. The uniqueness of g_t follows from the Doob-Meyer decomposition theorem. We shall give an alternative proof here as in Theorem 4.1. If Z_t = 1 + ∫_0^t g_s,i dB_s,   i=1,2, let h_t = g_t,1 - g_t,2 and J_t = ∫_0^t h_s dB_s. By hypothesis, J_t = 0 a.e. and we have to show the same for h_t. By the Itô formula,
		
    J_t^2 = ∫_0^t 2 J_s h_s dB_s + ∫_0^t h_s^2 ds,

		which leads to
		
    ∫_0^t h_s^2 ds = 0  a.e.
 and hence h_t = 0  a.e. 
		Returning to the main proof, we note that by applying the Itô formula to (<ref>), Z_t can be rewritten as
		
    Z_t = exp{∫_0^t g_s/Z_s dB_s - 1/2∫_0^t g_s^2/Z_s^2 ds}. ,    t < τ.

		But we can now obtain (<ref>) by letting ϕ_t = -g_t/Z_t, t < τ.
	
	
		The equivalence of (<ref>) and (<ref>) is assured by the Girsanov theorem.
	
	
	
	
	
	
	
	
	
	
	
	
	
	 If there exists 𝐏 with respect to which a Wiener additive process has the standard Brownian motion and 𝐏 || 𝐏_0, then it is non-anticipative in the sense of (<ref>) fulfilled. 
	
	
	Another example is the Markov process. The Markov additive process X_n with finite state space {1,2,…,N} is characterized by the transition probability P^(m) which is a N× N matrix such that 
	
		
  5.D X_n is adapted to ℱ_n
		
  5.E X_n has the Markov property with respect to ℱ_n
		
    𝐏(X_n+m = i|ℱ_n) = P^(m)_X_n,i,   ∀ i ∈{1,…,N}, n,m =1,2,…
  
	
	By the definition 5.E we abbreviate P^(m) = P^m where P = P^(1).
	
		If there exists 𝐏_0 with respect to which a Markov additive process X_n on {1,2,…,N} has the stationary Markov chain and 
		
    𝐏_0(X_n+m = i | X_n=j) = 0

		implies 
		
    𝐏(X_n+m = i | X_n=j) = 0

		for any i,j∈{1,…,N}, n,m= 0,1,2,…, then it is non-anticipative in the sense of (<ref>) fulfilled.
	
	
	
		Let X_n be a Markov additive process with σ(X_1,X_2, …, X_m)-transition probability matrix P^(m). Suppose 𝐏_0 is a probability measure on (Ω, ℱ_T) with respect to which X_n is the stationary Markov chain, then 𝐏≪𝐏_0 and
		
    .d𝐏/d𝐏_0|_σ(X_1,X_2,…, X_n)= exp[∑_i=1^nlog(P_X_i,X_i+1/P_0,X_i,X_i+1) ]

		Conversely, if there exists 𝐏_0 as described above and 𝐏≪𝐏_0, then there exists a σ(X_1,X_2,…,X_m)-transition probability matrix P^(m) such that (<ref>) holds. 
	
	 Sufficiency. Similar to Theorem 4.1. 
		
			In order that X_n is a Markov chain with transition probability P, it is necessay and sufficient that for any z=(z_1,…,z_N) ∈𝐑^N,
			
    z_X_n - z_X_0 - ∑_m=1^n(zP)_X_m

			is a σ(X_m: 0 ≤ m ≤ n)-martingale with probability distribution 𝐏v_0^-1 uniquely given by the initial distribution v_0 = (v_0,1,…,v_0,N) ∈𝐑^N and v_0,1+…+v_0,N = 1.
		
		 Omitted.
		
		
		Necessity. Therefore by Claim 5.7 any local martingale has the predictable representation property with respect to (<ref>). Suppose there exists a σ(X_1,X_2,…,X_n)-predictable process G_n satisfying for the 𝐏_0-martingale Z_n similar as above
		
    Z_n = 1 + (G · X^ε)_n

		where G · X = ∑_m=1^n G_m(X_m - X_m-1), ε = (1,2,…,N) and X^ε_n is given by (<ref>) with z = ε. To find P such that (<ref>) holds, 
		
    𝐄 [C · X] = 𝐄_0 [ (ZC)· X ] = 𝐄_0 [(ZC)· A]

		where A_n = ∑_m=1^n (z P_0)_X_m. Take 
    P_i,j = [1 - (1 + G_n/Z_n-1)P_0,i,j]/[(1 + G_n/Z_n-1) P_0,i,j]

		gives the desired result.
	
	
	

§ ACKNOWLEDGMENTS

	
	The author thanks professor Hui Huang for his guidance and associate professor Yuan Zhang for pointing out an error in the expression (<ref>) in the original manuscript. Also, thanks professor Zhi-ming Li for reminding me that two Wiener processes may not be equivalent unless they have equal variance.
	
	
	
	amsplain
	


 Paper Title: Infinitesimal behavior of Quadratically Regularized Optimal Transport and its relation with the Porous Medium Equation.


[
    [
    Received 2 August 2023 / Accepted 7 May 2024
================================================

§ ABSTRACT 
 The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be sparse, i.e., the density of the coupling needs to be zero for a large subset of the product of the supports of the marginals. However, unlike the acclaimed entropy-regularized optimal transport, the effect of quadratic regularization on the transport problem is not well understood from a mathematical standpoint. In this work, we take a first step towards its understanding. We prove that the difference between the cost of optimal transport and its regularized version multiplied by the ratio ε^-2/d+2
  converges to a nontrivial limit as the regularization parameter ε tends to 0. The proof  confirms a conjecture from <cit.> where it is claimed that a modification of the self-similar solution of the porous medium equation, the Barenblatt–Pattle solution, can be used as an approximate solution of the regularized transport cost for small values of ε.
  



§ INTRODUCTION AND MAIN RESULT


Optimal transport has an extremely rich history that started with Monge more than two centuries ago <cit.>. The problem is, given two  positive measures ρ_0 and ρ_1 with the same finite mass defined on Ω_0 and Ω_1 and a cost function from Ω_0×Ω_1 to [0, ∞), to find the optimal way of transporting one measure onto the other at the minimal overall cost. That is, the optimal transport cost between two probability measures having densities ρ_0 and ρ_1 with respect to the d-dimensional Lebesgue measure  ℓ_d is defined as 

    𝒲_2^2(ρ_0, ρ_1) = inf_π∈Π(ρ_0, ρ_1)∫-^2 π(,),

where Π(ρ_0, ρ_1) denotes the set of probability measures π∈𝒫(^d×^d) such that 

    ∫ f() π(, )= ∫ f() ρ_0()    and   ∫ f() dπ(, )= ∫ f() ρ_1() ,

for any bounded continuous function f∈𝒞_B(^d). In the case where ρ_1 is absolutely continuous with respect to the Lebesgue measure ℓ_d, the solution π̃ of (<ref>) is unique and of the form π̃ =( ∇ g× Id)_#ρ_1, where  g is a convex function (see <cit.>). In such a case, it is said that ∇ g pushes ρ_1 forward to ρ_0.       We refer to the two monographs by <cit.> for a detailed exposition.

We highlight that, in a common abuse of notation, we will refer—when ambiguity is absent—to the densities ρ_0 and ρ_1 and to the measures defined by them often by just speaking about the measures ρ_0 and ρ_1 or ρ_0 and ρ_1. 

After a series of fundamental theoretical developments, among which those of Kantorovich, <cit.> and many more, 
 the interest for optimal transport exploded in recent years as its usefulness for various application domains such as Machine Learning <cit.>, Statistics <cit.>, Biology <cit.>, Astronomy <cit.> or Economy <cit.>  got acknowledged. Note that it became impossible to list all the contributions to the field.

Moving away from the classical framework, a recent line of research has been inspired by new computational developments available for a particular relaxation of the optimal transport problem. The latter relaxation consists in  adding a (convex) penalty to the initial problem given in equation (<ref>) above. A particular such penalty, spearheaded by <cit.> for its easy computability on GPU, relies on the addition of a Kullback–Leibler regularization term
in the initial objective function, i.e., 

    𝒯_2, , KL(ρ_0, ρ_1)= inf_π∈Π(ρ_0, ρ_1)∫-^2dπ(,)+  KL(π| (ρ_0 ⊗ρ_1)),

where (ρ_0 ⊗ρ_1) denotes the product measure and KL(α|β)=∫log( α /β)α if α is absolutely continuous w.r.t. β  (i.e., α≪β) and KL(α|β)=+∞ otherwise. Here α /β stands for the Radon–Nikodym derivative of α w.r.t. β. Conversely, given a positive function ω we denote as ωβ the measure having Radon–Nikodym derivative ω w.r.t. β.
The addition of this penalty term  enables the use of a matrix-scaling algorithm: the Sinkhorn algorithm <cit.>. A plethora of works and applications followed, see <cit.>  for a book-long exposition.


Studying the infinitesimal behavior of the difference in the regularized cost  as the regularization parameter  tends to 0, <cit.> proved the limit 

    lim_→ 0^+𝒯_2, , KL(ρ_0, ρ_1)- 𝒲_2^2(ρ_0, ρ_1)- d/2 log(π)/= ∫log(ρ_0) dρ_0-∫log(ρ_1) dρ_1 /2

using a Gaussian approximation of the minimizers of the regularized problem. This is a natural approximation from a heuristic point of view, given that Gaussian distributions are the fundamental solutions of the heat equation, whose Wasserstein gradient flow is over the Kullback–Leibler entropy <cit.>.


Just like in the heat equation, whose  solution has maximum support for any positive time, the solutions of the optimal transport problem regularized by logarithmic entropy share the same property. That is, for every >0, the resulting coupling  is  dense, i.e., each point on the product space Ω_0 ×Ω_1 has a positive density. On the contrary, the (unregularized) optimal transport plan for the squared Euclidean distance as a cost is concentrated on the gradient of a convex function.   To remedy this, <cit.> introduced the Quadratically Regularized Optimal Transport (QOT)

    𝒯_2, ε, (·)^2(ρ_0, ρ_1) = inf_π∈Π(ρ_0, ρ_1)∫-^2π(,)+π/ (ρ_0 ⊗ρ_1) _L^2( ρ_0 ⊗ρ_1)^2,
  
where  

    π/ (ρ_0 ⊗ρ_1) _L^2( ρ_0 ⊗ρ_1)^2=∫( π/ (ρ_0 ⊗ρ_1))^2  (ρ_0 ⊗ρ_1)

in the case where π≪ (ρ_0 ⊗ρ_1), and +∞ otherwise. 

Applications of QOT have since then started blossoming. Examples include <cit.>. For some theoretical results we refer to <cit.>. <cit.> observed that the minimizer of (<ref>) is  “sparse” in the sense that its topological support seems to increase progressively as  gets larger. 
This increase of a sparse support suggests a diffusion behavior like that of the Porous Medium Equation (PME) where epsilon would play the role of the time parameter. For this reason, before setting this parallel on solid ground, we provide a short introduction to the PME in the next section.



 §.§ On the Porous Medium Equation and the QOT 


The Porous Medium Equation posed in the whole space reads, for an initial datum u_0,

    ∂_t u(t,) =Δ u(t,)^m,      t>0, ∈ℝ^d
    
        u(0,)=u_0()     x∈ℝ^d,

where m>1 is the non-linearity exponent. A mathematical study for this equation appeared first, in the case m=2, in the study of groundwater infiltration by <cit.> and since then it has attracted the attention of numerous experts. The general case m>1 can be derived from the study of an an-isotropic gas in a porous medium (hence the name) subject to Darcy's Law, see the seminal (and independent of each other) works of <cit.> and <cit.>. Since then, it has enjoyed a continuous attention from the mathematical community due to its applications in many fields, like plasma radiation, biology, population theory or even cosmology. We refer the interested reader to the book of <cit.>, broadly considered as the go-to reference on this matter. Therein, the reader can find a more detailed explanation of the main characteristics of the PME that are relevant for the study of the QOT. Even though in this article we focus on the case m=2, we will discuss the general case m>1.

The first property that leaps to the eye is the fact that the PME is non-linear, i.e., the sum of two solutions of the equation is not a solution itself. This makes it impossible for the existence of a linear description of its solutions similar to the heat equation, where the solution can be expressed as the convolution of the initial datum u_0 and the fundamental solution, the self-similar Gaussian profile (also referred as the kernel of the equation). The solution of the PME cannot be expressed as the convolution of the initial datum  with any kernel but, nevertheless, there exists a fundamental solution ℬ(t,x) of the PME, in the sense that

    ∂_t ℬ(t,) =Δℬ(t,)^m,      t>0, x∈ℝ^d
    ℬ(0,)=δ_0()    ∈ℝ^d,

where δ_0() is Dirac's Delta function located at =0 and the previous equalities should  be understood in the weak sense (or integral sense). This fundamental solution is called Barenblatt or Barenblatt–Kompaneets–Zeldovich profile due to its discoverers <cit.> and <cit.>. It reads

    ℬ(t,)=1/t^α[C- βm-1/2m^2/t^2β]_+^1/m-1,  where  α=d/d(m-1)+2, β= 1/d(m-1)+2,

and C>0 is a free constant that dictates the mass of the profile.
Contrarily to the Kullback–Leibler regularisation case, we cannot work with linear tools, i.e., semigroup theory. Still,  the profile of the solution above will play a key role in our work, see formula (<ref>) to mention but one example. 

The next property that is relevant in our study is the slow nature of the diffusion in the PME. If one writes the equation in terms of its diffusivity D(u) then we see that

    ∂_t u= div(D(u)·∇ u ),  with   D(u)=m u^m-1

and therefore the diffusivity disappears whenever u=0; in other words, our equation is degenerate parabolic at the points where u=0. This provokes the phenomena known as slow diffusion, where every solution whose initial datum has compact support will maintain a compact support that grows over time, in sharp contrast with the heat equation, where every solution becomes positive everywhere at any positive time. This characteristic of the PME relates closely to the sparse nature of the minimizer in the QOT observed by <cit.>.

Finally, there is one last connection between the PME and the QOT that we would like to mention. <cit.> explained how that the PME could be understood as a gradient flow with the 2-Wasserstein metric of the functional formed by the second moment of the solution and its free energy, namely

    E(u)=∫1/2^2u() + 1/m-1u()^m  dℓ_d(),

see also <cit.> for some applications. More or less at the same time <cit.> followed this approach, using E(u) as the natural entropy of the system, to describe the large-time behavior of the solutions of the problem. Note that  the resemblance between (<ref>) and  (<ref>) in the case m=2. Quadratically regularized optimal transport thus mimics the minimisation of the free energy just defined under the coupling constraints. As we shall see later, this last observation is where the geometry of unregularized optimal transport comes into the picture and plays a crucial role.



 §.§ Assumptions and main results


The existence of a potential link between QOT and the porous medium equation was considered folklore in the community in the last years even though never rigorously established. The first developments clearly fleshing this out were proposed by <cit.>. Their work further suggests that, by using Barenblatt profile approximations instead of Gaussian approximations (for the case of the Kullback–Leibler regularisation studied by <cit.>), one could characterize the non-trivial limit of 

    r()  (𝒯_2, ε, (·)^2(ρ_0, ρ_1) -𝒲_2^2(ρ_0, ρ_1)  )
   
for a proper choice of r(). Via a proof   based on quantization, <cit.> determined the value of the ratio r(ε) as r(ε) = ε^-2/d+2. It is worth highlighting, however, that the nonlinear nature of the penalty in (<ref>) makes impossible to utilize the linear tools present in <cit.> or most of the techniques present in the existing literature. The method we propose here is novel and relies mostly on the understanding of Barenblatt-like densities, inspired by (<ref>), and on the clever usage of mass transportation theory. 

In this work, we prove the complete (implicit) conjecture of <cit.> with this ratio under the following assumptions. 

We assume the following: 

    
  * Ω_1 and Ω_0 be open bounded sets such that 
    there exists δ_0∈ (0, π) and σ>0 such that for  every i∈{0,1} and ∈Ω_i there exists a cone
    
    𝒞_,  v,δ_0, σ={+ u:  v ucos(1/2θ) ≤⟨ v ,   u ⟩≤ vδ_0  }
 
    with vertex ,  height δ_0 and angle θ such that 𝒞_,  vδ_0, σ⊂Ω_i.
    
    
    
  * The functions ρ_0∈𝒞^β(Ω̅_̅0̅) and ρ_1∈𝒞^β(Ω̅_̅1̅) for some β>0 are densities defining  probability measures 
    
    1=∫_Ω_iρ_i() ℓ_d()=∫ dρ_i()=ρ_i(Ω),    i=1, 2.

    
  * Such  densities are upper lower bounded in their domain, i.e.,  

    λ≤ρ_0() ≤Λ  and λ≤ρ_1() ≤Λ  for all ∈Ω_0   and ∈Ω_1

for some 0< λ≤Λ < ∞.

  * The unique gradient of a convex function g pushing ρ_1 forward to ρ_0 belongs to 𝒞^2, α(Ω_1) and 

    σ_m(g)  Id≤∇^2 g ≤σ_M(g)  Id   in  Ω_1

for some 0<σ_m(g) ≤σ_M(g)< ∞.


Note that Assumption <ref>, 4. implies also that the convex conjugate of g, namely

    g^*()=sup_∈^d{⟨, ⟩ - g() },

is 𝒞^2(Ω_0). Moreover, it is well known that ∇ g^* pushes ρ_0 forward to ρ_1.  

The following is our main result. 




    Let Assumption <ref> hold and g be the unique function such that ∇ g pushes ρ_1 forward to ρ_0.   Then it holds that
    
    lim_ε→ 0^+𝒯_2, ε, (·)^2(ρ_0, ρ_1) -𝒲_2^2(ρ_0, ρ_1) /ε^2/d+2  = d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫_Ω_0(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0(),



 
 



As a consequence of this result, we can write explicitly the relationship between solutions of the PME and the optimizers of the QOT. Define the energy 
  
    E_(π):=   ∫1/2-^2 π(,)+ε·π^2(, )  (ρ_0 ⊗ρ_1)(,)

  and let g be the unique function such that ∇ g pushes ρ_1 forward to ρ_0. Define then, for any '∈Ω_0,
  
    ϱ('):=ρ_0( [∇^2g^*(')]^-1/2')·ρ_1(∇ g^*( [∇^2g^*(')]^-1/2')).



Choose now any '∈Ω_0 and let u(t, ; ') be the solution of 
  
    ∂_t u(t,;') =1/2(d+2)Δ_ u(t,; ' )^2,      t>0, ∈Ω_0
    
        u(0,; ')=ϱ(')^-1/d+2·δ_'()    ∈ℝ^d,

where Δ_ represents the Laplacian only in the 's variable, δ_'() denotes the Dirac's Delta function on the point =' and the previous solution must be understood in the weak sense. The constant 1/2(d+2) in front of the Laplacian operator can be considered just as a scaling factor in the  variable. The formula for u reads

    u(t,;')=1/t(C_d  t^2/d+2/ϱ(')^1/d+2 -1/2-'^2 )_+

where C_d is just a normalizing constant depending only on the dimension of the space that will be stated later.

Next, to each '∈Ω_0 and each solution u of the previous PME it corresponds a transformed function
  
    v(t,;')   := u(t, [∇^2g^*(')]^1/2; [∇^2g^*(')]^1/2')
       = 1/t(C_d  t^2/d+2/(ρ_0(')ρ_1(∇ g^*(')))^1/d+2 -1/2-'^2_∇^2 g^*('))_+,

  where
  
    -' ^2_∇^2 g()= ⟨-',  ∇^2 g()(-')⟩,
 
We are ready now to state the corollary.

    Under the assumptions of Theorem <ref> let π_() be the minimizer of (<ref>) for >0 and. Then it holds 

    lim_ε→ 0^+ε^-2/2+d| E_(π_()(,))-E_(v(, , ∇ g()))|=0




We sketch a proof of the corollary for the interested reader. The details are omitted, as they can be readily deduced by repeating the  calculations provided in the proof of Theorem <ref>.  Later on we will define a family of approximations π_ε to the true minimizer π_(ε). The key of these approximations is that, as ε→ 0,

    π_ε(,)∼ v(ε,;∇ g()).

The idea consists then in writing, adding and subtracting terms,

    ε^-2/2+d[ E_(π_()(,))-E_(v(, , ∇ g()))]    =ε^-2/2+d[ E_(π_()(,))- 𝒲^2_2(ρ_0,ρ_1)]_ℐ
       + ε^-2/2+d[ 𝒲^2_2(ρ_0,ρ_1)-E_(π_(,))]_ℐℐ
       +ε^-2/2+d[ E_(π_(,))-E_(v(, , ∇ g()))]_ℐℐℐ.

Then thanks to Theorem <ref> and the fact that π_ε approximates π_(ε) we have ℐ→ -ℐℐ, and then from (<ref>), applying the same ideas as in the proof of Theorem <ref>, we obtain that ℐℐℐ→ 0.




Define v(, , ')=u(, [∇^2 g^*()]^1/2, [∇^2 g^*()]^1/2'))
si 
    u(, , ')=1/( C_(')- 1/2-'^2 )_+

    
    E_(u(, ·, ∇ g(·)))= 1/2∫-^2 u(, , ∇ g())+ε/2 u(, , ∇ g())^2  (ρ_0 ⊗ρ_1)(,)
    
            =1/2∫-∇ g^*()^2 u(, , ')+ε/2 u(, , ')^2  (ρ_0 ⊗ρ_0)(,')




Apart from obtaining the limit conjectured by <cit.>, the main interest of this article consists on providing a constructive approach to the study of such limits for more general penalties. Our technique should work with very little changes for penalties of the form ·_L^m( ρ_0 ⊗ρ_1)^m for m>1, which corresponds to the full range of parameters of the Porous Medium Equation. We leave this generalization for further work and we highlight that we do not know if this technique could work for non-convex penalties of the form ·_L^m( ρ_0 ⊗ρ_1)^m for 0<m<1. This approach differs greatly from the one taken by <cit.>, which relies deeply on the linearity of the heat equation.   In addition, our analysis further  elucidates why the correct rate of convergence in Theorem <ref> is ^-2/d+2.  





To prove Theorem <ref>, we need to provide both an upper bound and a lower bound for 𝒯_2, ε, (·)^2(ρ_0, ρ_1). To provide the lower bound, we will use the dual formulation (see <cit.>)

    𝒯_2, ε, (·)^2(ρ_0, ρ_1) =2·sup_a,b∈ L^2(ρ_0)× L^2(ρ_1)∫{a() + b() 
    
        -1/2(a()+b()-1/2-^2)_+^2}ρ_0() ρ_1()

of (<ref>), whose solutions  are the so-called regularized potentials. 
The solutions of the primal and dual problems are related in the following way;  (a_, b_) is a maximizer of (<ref>) if and only if 

    1/(a_()+b_()-1/2-^2  )_+ρ_0() ρ_1()

is a minimizer of (<ref>). The reader should note the resemblance between (<ref>) and (<ref>) in the case m=2, with a_ε(), b_ε()∼ Cε^2/d+2 and t=ε. This relation suggests as a candidate  a pair of functions  (f̃_, g) such that 

    ∫(f̃_()+1/2^2 -g()-1/2-^2 )_+ dρ_1()=     for all ∈Ω_0.

After some calculations, 
Lemma <ref> yields 
f̃_()≍1/2^2 -g^*()+ C_(),
where 
    C_():=^2/d+2/C_d^2/d+2(ρ_0()ρ_1[∇ g^*()])^1/d+2   for    C_d:= 2^d+2/2ℋ^d-1(𝒮^d-1) 1/d(d+2).

Hence, we will find the lower bound by finding the limit as → 0 of 

    2·^-2/d+2(Γ_(f_, 1/2·^2 -g)- 1/2𝒲_2^2(ρ_0, ρ_1))

where 
    f_()=  1/2^2 -g^*()+ C_() and 
    
    Γ_ (a, b)=∫{a() + b() 
        -1/2ε(a()+b()-1/2-^2)_+^2}ρ_0() ρ_1().

For the upper bound, we will find a coupling π_∈Π(ρ_0, ρ_1) such that the functional 

    ℍ_(π)=   ∫-^2dπ(,)+dπ/  d(ρ_0 ⊗ρ_1) _L^2( ρ_0 ⊗ρ_1)^2,

at π_ achieves the same limit. The natural candidate is the measure π̃ with  π̃/ (ρ_0⊗ρ_1) (, ) = 1/(C_()-D(, ))_+, 
where 

    D(,  )= g^*()+g()-⟨, ⟩

is the  well-known  Bregman divergence. 
However, π̃ is neither a coupling nor a probability measure in the product space. 
Therefore, it must be adjusted using the following procedure: Firstly, transport ρ_0 to ρ_1 to work exclusively with the measure

    1/ε(C_ε(𝐱) - D(𝐱, ∇ g^*(𝐱')))_+ dρ_0(𝐱)  dρ_0(𝐱').

Since D(𝐱, ∇ g(𝐱')) is more difficult to control than its quadratic approximation (see Lemma <ref>)  
    1/2-' ^2_∇^2 g()=1/2⟨-',  ∇^2 g()(-')⟩,
 
we construct the symmetric version of its density

    m_ε (𝐱, 𝐱') = 1/2 ε( C_ε(𝐱) + C_ε(𝐱') - 1/2-' ^2_∇^2 g()-1/2-' ^2_∇^2 g('))_+.

Again, we highlight the resemblance between this last definition and (<ref>) in the case m=2, with C_ε(), C_ε()∼ Cε^2/d+2, also -' _∇^2 g(),-' _∇^2 g(')∼-' (see Lemma <ref> later on) and finally t=ε.

Subsequently, we normalize to create the measure with density 

    ξ_ε (𝐱, 𝐱') = m_ε(𝐱, 𝐱')/∫ m_ε(𝐱, 𝐱')  d(ρ_0 ⊗ρ_0)(𝐱, 𝐱')
 
with respect to ρ_0 ⊗ρ_0.
Within the product space, the symmetry ξ_ε is important because it implies   that both its marginals are the same and with  density ρ_(')=∫ξ_ (, ') ρ_0() with respect to ρ_0. Then, transport ρ_ερ_0 to ρ_0 and define the probability measure 

    dμ_ε (𝐱, 𝐲) = ξ_ε(∇ϕ_ε^*(𝐱), ∇ϕ_ε^*(𝐱'))/ρ_ε(∇ϕ_ε^*(𝐱)) ρ_ε(∇ϕ_ε^*(𝐱'))dρ_0(𝐱)  dρ_0(𝐱'),

where ∇ϕ_ε is the optimal transport map between  ρ_0 and ρ_ερ_0. For the rates of this coupling to be appropriate, it is necessary that the Lipschitz constant of ∇ϕ_ε tends to 1 uniformly (see (<ref>)). Using Caffarelli's regularity theory (see <cit.>), one can obtain the norm ϕ_ε_𝒞^2, α(Ω_0^δ), where 

    Ω_0^δ = {𝐱∈Ω_0 : dist(𝐱, ∂Ω_0) ≥δ}
 
and δ > 0, uniformly bounded in ε. In one dimension or in the flat torus, this holds for δ = 0, and the proof concludes at this step. In higher dimensions and with Euclidean cost, it is not entirely clear that this bound has been established in the literature—regularity up to the boundary is known for strongly convex and smooth domains  <cit.>, but the dependence on the constants is not clear to us. The same issue appears in <cit.> in a different context. Instead of attempting to prove this result, we will circumvent the issue by dividing the space Ω_0 into two parts: points that are less than δ away from the boundary and those that are farther. For the latter, we can apply interior regularity, where the dependence on the constants is explicit. For points near the boundary, we will use a different coupling—the so-called “frame coupling”—that does not achieve the appropriate limit but does not cause the functional to diverge. After taking the limit as   tends to 0, we will then take the limit as δ tends to 0 and check that it matches the lower bound.

As Caffarelli's results are only valid for convex domains, the previous discussion is applicable only to convex domains. 
The final step of the proof is to remove that assumption. It involves creating a tessellation of the domain Ω_0 into disjoint squares {I_i}_i=1^m and constructing a frame coupling in Ω_0 ∖{I_i}_i=1^m, applying the previous coupling to each of the squares. This piecewise-defined coupling is referred to as the “stained glass" coupling (see Figure <ref>).

The rest of the paper is organized as follows. In Section <ref>, we introduce the notation that will be utilized throughout the paper.  In Section <ref>, we present the necessary results from optimal transport theory that are essential for proving the main theorem. In Section <ref>, we will provide the lower bound, and in Section <ref>, we will establish the upper bound. As mentioned earlier, the upper bound will be proven first for convex supports (Section <ref>) and then for arbitrary supports (Section <ref>). 






§ NOTATION AND AUXILIARY RESULTS

In order to present the foundational concepts more seamlessly, we begin by defining a finite positive constant C(a_1, …, a_d) that depends solely on a set of parameters (a_1, …, a_d). For the sake of brevity, we denote a constant that depends only on ρ_0, ρ_1, Ω_0, Ω_1 and g by C, specifically C = C(ρ_0, ρ_1, g). A function v of ε is said to be a little-o of another function u (denoted as v = o(u)) if v(ε)/u(ε)→ 0 as ε→ 0. Occasionally, the function v(ε, 𝐱) may depend on 𝐱 within Ω_0. In such cases, we use the notation v = o(u) if sup_𝐱∈Ω_0v(ε, 𝐱)/u(ε)→ 0 as ε approaches zero. In the same manner, we will use the standard big-O notation.
 The notation x ≍_κ y is used to express an equivalence of order of convergence, considering multiplicative constants that depend solely on a parameter κ > 0, particularly in contexts approaching the limit as ε→ 0. As before κ=(ρ_0, ρ_1, Ω_0,
Ω_1, g) we just write x ≍  y.

Moreover, within a compact set Ω, the space 𝒞^p, α(Ω) is the space of Hölder continuous functions of order p with Hölder continuity parameter α in the open interval (0,1). The norm for this function space is defined as 

    f_𝒞^p, α(Ω) = ∑_k=0^p∑_|α|=ksup_x ∈Ω| d^k/dx_a_1⋯ dx_a_k f(x) | + ∑_|α|=psup_x, y ∈Ω
     x ≠ y|f(x) - f(y)|/x - y^α.


Additionally, the notation ℓ_d is employed to denote the Lebesgue measure, and g^* refers to the convex conjugate of g, calculated as g^*(x) = sup_y ∈ℝ^d (⟨ x, y ⟩ - g(y)). The Hausdorff measure of dimension d-1 is represented by ℋ^d-1.









Furthermore, for a positive definite matrix A, its square root, denoted A^1/2, is the unique positive definite matrix V such that V^2 = A.  The product measure ρ_0 ⊗ρ_1 and the space L^2(μ), which includes functions f with a finite L^2(μ)-norm given by

    f_L^2(μ) = ( ∫ f^2(x)  μ(x) )^1/2,

are also crucial. In instances where Borel measures μ and ν satisfy μ≪ν, the notation dμ/dν represents the Radon-Nikodym derivative of μ with respect to ν.  For the particular cases of ρ_i, for i=1,2, to avoid complicating the notation, we will use the following abuses of notation: dρ_i/dℓ_d= ρ_i and 

    ∫ f() dρ_i() = ∫_Ω_i f() ρ_i()  ℓ_d(),    i=1, 2.

For a Borel set A,   χ_A denotes the indicator function of A. Sometimes, when it can cause confusion, we write χ_[A]. We say that a function T pushes a measure μ forward to another measure ν if ν(A)=μ(T^-1(A)) for all Borel set A. In such a case, we write T_#μ=ν.  A set A is compactly contained in B (A⊂⊂ B)  if there exists a compact set K and an open set 𝒰 such that A⊂ K⊂𝒰⊂  B. The Euclidean Ball with center  and radius δ > 0 is denoted as 𝔹(, δ).






 §.§ Optimal transport background

The optimal transport problem  (<ref>) admits the dual formulation

    𝒲_2^2(ρ_0, ρ_1) 
       =2·sup_(a,b)∈Φ∫ a() ρ_0() + ∫ b() ρ_1(),

where 
Φ={ (a, b)∈ (𝒞(^d))^2:  a()+b()≤-^2/2,  ∀, ∈^d} .
Under Assumption <ref>, the solution of the primal problem (<ref>) is given by ( Id×∇ g)_#ρ_1 and the solution of (<ref>) by the pair 

    ( 1/2·^2 -g^*,  1/2·^2 -g ).

In general, the function g solves the Monge-Ampère equation

    (∇^2 g)=ρ_1/ρ_0(∇ g)    in Ω_1,   ∇ g (Ω_1)= Ω_0

in the so-called Brenier sense (see <cit.>). In our case, by assumption, g solves it in the strong sense. The Monge-Ampère equation is a degenerate elliptical nonlinear differential equation. The regularity of convex solutions has been studied by Caffarelli in <cit.>. Here, we highlight the result that we will use in this work.
[Caffarelli]
   Let Ω be open and convex q_1,q_2∈𝒞^0,α(Ω) be positive functions such that 
    
    ∫_Ω q_1  dℓ_d= ∫_Ω q_2   dℓ_d=1.

    Then there  exists a
    unique (up to additive constant) convex solution u of the boundary value problem 
    
    (∇^2 u  )=q_1/q_2(∇ u )   in Ω,   ∇ u (Ω)= Ω

    in 𝒞^2,α_Loc(Ω).
    Moreover, for each K⊂⊂Ω 
    
    u_𝒞^2, α(K)≤  C(K, inf_∈Ω q_1, inf_∈Ω q_2, q_1_𝒞^0,α(K), q_2_𝒞^0,α(K))


The assumption of regularity on g is key in the infinitesimal developments we are undertaking. Lemma <ref> is fundamental as it relates D(, ) with its linearized inner product 
 
    ⟨, ⟩_∇^2 g^*()= ⟨, ∇^2 g^*() ⟩.
 
 We omit its proof as it is a mere second-order Taylor expansion of the Bregman divergence 

    D(, )= g^*()+g()-⟨, ⟩

of g.


    Let Assumption <ref> hold. Then there exists σ_M(g)≥σ_m(g)>0 such that 
    σ_m(g)/2-'^2   ≤  D(, ∇ g^*('))≤σ_M(g)/2-'^2.
 
Moreover,  there exists C>0 such that 

    |    D(, ∇ g^*('))-1/2-' ^2_∇^2 g^*()|≤  C-'^2+α,
   
where 

    1/2-' ^2_∇^2 g^*()=1/2⟨-',  ∇^2 g^*()(-')⟩

and α is as in Assumption <ref>. 


As a direct consequence of Lemma <ref> we obtain the following.

 
     Let Assumption <ref> hold. Then the following estimates hold: 
     
         
  * There exists a constant C and _0>0  such that  
    
    { (, ')∈Ω_0×Ω_0:     C_()≥  D(, ∇ g^*('))  }⊂{ (, ')∈^2d:  - '≤ C^1/d+2}

        holds for every 
         ≤_0. 
         
  * For ≤_0, 
     
    sup_, '∈Ω_0|(  C_()- D(, ∇ g^*(')))_+
            - (  C_() -1/2-'_∇^2 g^*()^2)_+|
            =o(^2/d+2).

       
 



 §.§ Properties of the integrals


The following results are well-known and easy to prove passing to spherical coordinates. We state them for further reference.


For any a≥0,    it holds that 
    
    ∫( a- ^2/2)_+ ℓ_d()̆= |a|^d+2/2 C_d^(1)  ,

    where 
    
    C_d^(1) := 2^d+2/2/d(d+2)ℋ^d-1(𝒮^d-1),

    and also
    
    ∫(a- ^2/2)_+^2 ℓ_d()̆= |a|^d+4/2 C_d^(2),

    where 
    
    C_d^(2):= 2^d+6/2/d(d+2)(d+4)ℋ^d-1(𝒮^d-1).








§ LOWER  BOUND

The aim of this section is to show 

    lim inf_→ 0^+𝒯_2, ε, (·)^2(ρ_0, ρ_1) - 𝒲_2^2(ρ_0, ρ_1)/^2/d+2≥d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().

A sufficient condition for this to occur is given by the following result —see equation (<ref>) in the introduction, whose proof will be provided below. 
One of the key aspects of this proof is that we divide the set Ω_0 into two, the points near its boundary or far away, namely

    Ω_0^δ := {∈Ω_0:   dist  (, ∂Ω_0)> δ}   and   Ω_0^δ,c:={∈Ω_0:   dist  (, ∂Ω_0)≤δ}.

These sets will also play a fundamental role in the proofs of Section <ref>, hence their importance. As the reader will see, being at a distance δ from the boundary will provide many useful uniform estimates that are in the core of our analysis. Then, making δ→ 0 we will find that our results hold true. 



    Let Assumption <ref> hold.  Set C_d:= 2^d+2/2ℋ^d-1(𝒮^d-1) 1/d(d+2). Then 
   
    
    lim_→ 0^+Γ_(f_, 1/2·^2 -g )- 1/2𝒲_2^2(ρ_0, ρ_1)/^2/d+2=d/C_d^2/d+2∫(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0(),

    where 
    f_()=  1/2^2 -g^*()+ C_() for C_()=^2/d+2C_d^-2/d+2(ρ_0()ρ_1[∇ g^*()])^-1/d+2.



As 
 
    Γ_(f_, 1/2·^2 -g )=  ∬{1/2^2 -g^*()+ C_() + 1/2^2 -g^*()
       
     -1/2( C_() -g()-g^*()+1/2^2+1/2^2-1/2-^2)_+^2}ρ_0() ρ_1()

and, by (<ref>),
 
    𝒲_2^2(ρ_0, ρ_1)=∫(1/2·^2-g^*)ρ_0+∫(1/2·^2-g)ρ_1,

 the value of 
    L=lim_→ 0^+Γ_(f_, 1/2·^2 -g )- 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
 is given by the limit (multiplied by ^-2/d+2) of
 
 
    L_   =∫ C_() ρ_0() -1/21/∬(  C_()- D(, ∇ g^*(')))_+^2 ρ_0()ρ_0(')_L'_.

Since  
 
    ^-2/d+2∫ C_() ρ_0() = ∫1/C_d^2/d+2(ρ_0()ρ_1[∇ g^*()])^1/d+2 dρ_0(),

 we only need to deal with the limit of ^-2/d+2 L'_.
 
 Set now δ>0 and define the sets Ω_0^δ and Ω_0^δ,c as in (<ref>). Then, due to Lemma <ref>, there exists _δ such that for every ≤_δ, the set Ω_0^δ, c contains the set

    𝕍_={∈Ω_0: ∃ '∈∂Ω_0:   C_()≥  D(, ∇ g^*(')) }.

 Therefore, we split 
^-2/d+2 L'_ into two terms; 

    𝕋_1=^-d+4/d+2∫_Ω_0^δ∫(  C_()- D(, ∇ g^*(')))_+^2 ρ_0(') ρ_0()

and 

    𝕋_2= ^-d+4/d+2∫_Ω_0^δ,c∫(  C_()- D(, ∇ g^*(')))_+^2 ρ_0(')ρ_0(),

and estimate each one separately. 

Estimate for (<ref>).   Since  ρ_0 is uniformly continuous over Ω_0 and the support of the function (  C_()- D(, ∇ g^*(')))_+ concentrates uniformly on the diagonal =' (see Lemma <ref>), it holds that  

    𝕋_1= ^-d+4/d+2∫_Ω_0^δ∫_Ω_0(  C_()- D(, ∇ g^*(')))_+^2 ℓ_d(') (ρ_0()+o(1 )) ρ_0(').
 
Considering that Ω_0^δ and 𝕍_ are disjoint, it follows that the inner integral can be taken over ^d, i.e., 

    𝕋_1= ^-d+4/d+2∫_Ω_0^δ∫_^d(  C_()- D(, ∇ g^*(')))_+^2 ℓ_d(') (ρ_0()+o(1 )) ρ_0(').
 
Lemma <ref> implies 

    𝕋_1≍^-d+4/d+2∫_Ω_0^δ∫_Ω_0(  C_()-1/2⟨-', ∇^2 g^*() (-') ⟩ ))_+^2 ℓ_d(') ρ_0() ρ_0().
 
We focus on the inner integral

    I()= ρ_0^2() ∫_Ω_0(  C_()-1/2⟨-', ∇^2 g^*() (-') ⟩ ))_+^2 ℓ_d(') .
 First we change variables =̆  [∇^2 g^*()]^1/2 (-'), to get 

    I()=  ρ_0^2()/(∇^2 g^*())^1/2∫(  C_()-1/2^2)_+^2 
        ℓ_d()̆.

Using Proposition <ref>, we find

    I()   =  (C_())^d+4/2 C_d^(2)ρ_0^2()/(∇^2 g^*())^1/2
       =  ^d+4/d+2 C_d^(2)ρ_0^2()/C_d^d+4/d+2(ρ_0()ρ_1[∇ g^*()])^d+4/2(d+2)(∇^2 g^*())^1/2
       = ^d+4/d+2 C_d^(2)(ρ_0() ρ_1[∇ g^*()])^1/2/C_d^d+4/d+2(ρ_0()ρ_1[∇ g^*()])^d+4/2(d+2)ρ_0()
       = ^d+4/d+2 C_d^(2)/C_d^d+4/d+2(ρ_0()ρ_1[∇ g^*()])^1/(d+2)ρ_0(),

    where 
    C_d^(2):= 2^d+6/2/d(d+2)(d+4)ℋ^d-1(𝒮^d-1).
Hence, 

    𝕋_1 ≍  C_d^(2)/C_d^d+4/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().

We simplify the constant C_d^(2) C_d^-d+4/d+2 as follows:

    C_d^(2) C_d^-1 -2/d+2   = C_d^ -2/d+22^d+6/2/d(d+2)(d+4)ℋ^d-1(𝒮^d-1) 2^-d+2/2(ℋ^d-1(𝒮^d-1))^-1 d(d+2) 
       = 4/d+4C_d^ -2/d+2,

which leads to

    𝕋_1 ≍4/d+4C_d^ -2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().

Estimate for (<ref>). First note that under Assumption <ref>, 

    C_()≤^2/d+2/ C_d^2/d+2λ^2/d+2 .

Hence, due to Lemma <ref> and the fourth point of Assumption <ref>, we have the bound

    0≤   𝕋_2≤^-d+4/d+2∫_Ω_0^δ, c∫( ^2/d+2/C_d^2/d+2λ^2/d+2 -c-'^2)_+^2 ρ_0() ρ_0(')
       ≤Λ ^-d+4/d+2∫_Ω_0^δ, c∫( ^2/d+2/C_d^2/d+2λ^2/d+2 -c-'^2)_+^2 ℓ_d()  ρ_0(')
       ≤Λ/λ^2/d+2^-d/d+2∫_Ω_0^δ, cℓ_d({:̆ c^2≤^2/d+2/C_d^2/d+2λ^2/d+2})  ρ_0('),

for some positive constant c. As a consequence, there exists a constant c' such that 

    0≤𝕋_2≤ c' ρ_0(Ω_0^δ, c).

We can now conclude the proof by noticing that

    lim inf_→ 0^+    Γ_(f_, g)- 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
       ≥d/C_d^2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0() -Cρ_0(Ω_0^δ, c)

and 

    lim sup_→ 0^+   Γ_(f_, g)- 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
       ≤d/C_d^2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0() +Cρ_0(Ω_0^δ, c)

hold for any δ>0. Letting δ→ 0 we obtain the result.








§ UPPER BOUND


In this section, we establish an upper bound that is asymptotically equivalent to the lower bound.

The proof is quite technical. We will proceed as shown in Figure <ref>. First, we will assume that the set Ω_0 is convex and we will divide it between its interior Ω_0^δ and the points close to the boundary, Ω_0^δ,c. One can imagine Ω_0 as being a window where Ω_0^δ is its glass and Ω_0^δ,c its frame of width δ. In the glass part, we will define a coupling that achieves the appropriate limit, while in the frame, we will create one that tends to 0 as we let → 0 first and then  δ→ 0. For the general case, we will cover most of Ω_0 by composing a sort of stained-glass window formed by convex windows Ω_0,1, Ω_0,2, etc.... In each of the windows contained within the interior of Ω_0, we will use the same coupling as before. In those touching the boundary of Ω_0, we will use a coupling that does not blow up and decreases as the number of windows increases, albeit using many of smaller size.

Now we will present some technical results for bounding frame couplings in both the convex and general Ω_0 cases.




Let ρ_0 and g be as in Assumption <ref>. Let s:Ω_0↦ [a,b] for some 0<a<b<∞ be a given function and Ω⊂Ω_0 be any open set such that 
    there exists r_0>0 and θ∈ (0,π) such that for  every i∈{1, 2} and ∈Ω there exists a cone
    
    𝒞_,  v,r_0, σ={+ u:  v ucos(1/2θ) ≤⟨ v ,   u ⟩≤ vr_0  }
 
    with vertex ,  height r_0 and angle θ such that 𝒞_,  v,r_0, σ⊂Ω.
Then there exists  _0 and C=C(c,a,b)
such that for ≤_0 the following holds:

    
  *  There exists a function 
ψ_ such that

    s()=1/∫_Ω (ψ_ ()-D(, ∇ g^*(') )_+ρ_0(') ,   for all ∈Ω.


  *  There exists _0 and C>0 such that for every  ∈Ω and ≤_0, 

    ε^2/d+2/C≤ψ_() ≤ε^2/d+2 C,

and 

    ε^d/d+2/C≤ℓ_d(𝒮_ε, )≤ Cε^d/d+2,

where 𝒮_ε, = {'∈Ω_0:  ψ_ ()-D(, ∇ g^*(')  ≥ 0}.

  *  For every  '∈Ω, there exists a c'>0 such that

    1/∫_Ω (ψ_ ()-D(, ∇ g^*(') )_+ρ_0() ≥ c'.

 


 We prove each point separately. 

Proof of <ref>.
Set ∈Ω_0 and 

    Θ(a)=1/2 ∫_Ω (a-D(, ∇ g^*(') )_+^2ρ_0(')-a s().

Since s()>0, we have lim_a→±∞Θ(a)=+∞. Moreover, as 
Θ is   convex and differentiable,  there exists a minimizer a() which satisfies 

    1/ ∫_Ω (a()-D(, ∇ g^*(') )_+ρ_0(')= s() .

This proves the existence. 

Proof of <ref>.Again, since s()>0 for all ∈Ω_0,  ψ_  is non-negative  and Lipschitz with constant 2· diam(Ω)  (see the proof of <cit.>). This implies that 
ψ_   tends uniformly to 0 in Ω as → 0. 
Lemma <ref> implies that

    s()
           ≤∫_Ω(ψ_()- σ_m(g)/2-'^2)_+ ρ_0(')
       ≤Λ∫_Ω(ψ_()- σ_m(g)/2-'^2)_+ ℓ_d(')
       ≤Λ∫_^d(ψ_()- σ_m(g)/2-'^2)_+ ℓ_d(')
       ≤ C  (ψ_())^d+2/2,

for a certain constant C>0. Hence, the lower bound in (<ref>) follows.  Next, 

    s()    ≥∫_Ω (ψ_()- σ_M(g)/2-'^2)_+ ρ_0(')
       ≥λ∫_Ω(ψ_()- σ_M(g)/2-'^2)_+ℓ_d(').

By assumption, Ω contains the cone 𝒞_,  v,r_0, θ. As a consequence, 

    s()    ≥λ∫_𝒞_,  v,r_0, θ(ψ_()- σ_M(g)/2-'^2)_+ℓ_d(')
       =λ∫_𝒞_,  v,r_0, θ∩𝔹(,  (2ψ_()/σ_M(g))^1/2){ψ_()- σ_M(g)/2-'^2}ℓ_d(').

 Since  ψ_   tends uniformly to 0 in Ω, we can find _0>0 such that 
    ψ__𝒞(Ω)≤(2 r_0/σ_M(g))^2
 for all ≤_0. This yields 
 
    s() ≥λ∫_𝒞_,  v,ψ_(), θ∩𝔹(,  (2ψ_()/σ_M(g))^1/2){ψ_()- σ_M(g)/2-'^2}ℓ_d(').

Hence, there exists a constant C(θ,d, σ_M(g))>0 depending only on d, σ_M(g) and the ℋ^d-1-measure of the sector 

    𝒮^d-1∩{ u:  e_1 ucos(1/2θ) ≤⟨ e_1 ,   u ⟩},

where e_1 is the first element of the canonical basis of ^d,  such that 

    s()  ≥ C(d, θ) λ( ψ_())^d+2/2.

This gives the upper bound in (<ref>).  From here it is easy to obtain 
(<ref>) via the nesting

    𝔹(, 1/Cψ_())⊂𝒮_ε, ⊂𝔹(, C ψ_()),

which are consequence of Lemma <ref>.

Proof of <ref>. 
By the previous points, There exist _0 and c> 0
such that for ≤_0 

    ∫_Ω (ψ_ ()-D(, ∇ g^*(') )_+ dρ_0()   ≥∫_Ω(c^2/d+2- D(, ∇ g^*(')))_+ dρ_0() 
       ≥∫_Ω(c/Cψ_ (')- D(, ∇ g^*(')))_+ dρ_0()
    (by Lemma <ref>)    ≥ c_1 ∫_Ω(ψ_ (')- D(', ∇ g^*()))_+ dρ_0()
       ≥ c_1   inf_s().

Defining c':=c_1  inf_s() we obtain the desired result.





 §.§ Feasible coupling for the interior Ω_0^δ of a convex Ω_0


In this section we will define a possible coupling for the set Ω_0^δ. We will do so, in fact, by finding a coupling  μ_∈Π(ρ_0, ρ_0) that attains the correct limit when restricted  to (Ω_0^δ×Ω_0)∪ (Ω_0×Ω_0^δ) for δ>0. Then, we will restrict this coupling only to (Ω_0^δ×Ω_0)∪ (Ω_0×Ω_0^δ), but by doing so we will lose the part of the mass corresponding with Ω_0^δ×Ω_0^δ, which we will compensate with an extra term; see Section <ref> later on. 

We define, for ,'∈Ω_0×Ω_0,

    m_ε ( x,  x')=1/2 (C_()+C_(')-1/2-'_∇^2 g^*() ^2- 1/2-'_∇^2 g^*(') ^2 )_+ .

The following result is a direct consequence of Lemma <ref>, the definition of C_, the uniform continuity of ρ_0, ρ_1, and the transport map ∇ g, combined with the strict positivity of both ρ_0 and ρ_1. 

  Let Assumption <ref> hold. Then there exists C such that 
  
    -'≤ C ^2/d+2 ,

  for all (, ')∈ supp(m_)
and 
  
    sup_, '∈Ω_0|  m_ε ( x,  x')-1/(C_()- 1/2-'_∇^2 g^*()^2 )_+| =o(^2/d+2).


We define the 
function 

    ξ_ (, )=m_( , ')/∫ m_( , ')  (ρ_0⊗ρ_0)(, '),

which is a density (it is positive and integrates one) with respect to ρ_0⊗ρ_0. We define also 

    ρ_()=∫ξ_ (, ') ρ_0(')

as the marginal density (w.r.t. ρ_0), which by symmetry agrees with 
ρ_(')=∫ξ_ (, ') ρ_0().
Moreover it satisfies the following properties. We recall the definition of Ω_0^δ in (<ref>).















 Let Assumption <ref> hold. Then there exist two positive constants c ≤ C∈ (0, ∞) and a delta_0>0 such that for every δ<δ_0, there exists an _δ such that:


  *  ρ_ ρ_0 ∈𝒫(Ω_0) and  ξ_  (ρ_0 ⊗ρ_0)∈Π(ρ_ρ_0, ρ_ρ_0), 

  *   
    c ≤ρ_≤  C in  Ω_0,

  *   ρ_∈𝒞^0, α(Ω_0^δ) with 
ρ__𝒞^0, α(Ω_0^δ)≤ C'   for some positive constant C', and

  *  as → 0, 
    
    ρ_ -1_𝒞(Ω_0^δ)→ 0    and  ∫ m_ (·, ') ρ_0(') -1_𝒞(Ω_0^δ)→ 0 .



 Point <ref> holds by construction. We prove each of the remaining points separately. 

Proof of <ref> By definition of ξ_, it is enough to show the existence of two positive constants, c and C such that 
c ≤∫ m_(, ') ρ_0() ≤ C
for all ∈Ω_0. 
Set , '∈Ω_0.
 By Assumption <ref>, we have 
 
    C_()≤^2/d+2/C_d^2/d+2λ^2/d+2   and  1/2-'_∇^2 g^*() ^2 ≥σ_m(g)/2-'^2 .

As a consequence, 

    ∫ m_(, ') ρ_0()   ≤∫1/( ^2/d+2/C_d^2/d+2λ^2/d+2 -σ_m(g)/2-'^2 )_+ ρ_0()
       ≤Λ∫1/( ^2/d+2/C_d^2/d+2λ^2/d+2 -σ_m(g)/2-'^2 )_+ ℓ_d()
       =Λ∫1/( C_d ^2/d+2/C_d^2/d+2λ^2/d+2 -σ_m(g)/2^2 )_+ ℓ_d().

The latter quantity, call it C, does not depend on  and is finite, see Proposition <ref>.   The reverse inequality 

    ∫ m_(, ') ρ_0() ≥λ∫_Ω_01/( ^2/d+2/C_d^2/d+2Λ^2/d+2 -σ_M(g)/2-'^2 )_+ ℓ_d()

holds by the same means. From here, we conclude as in the proof  Lemma <ref>, claim <ref>. 


Proof of <ref>. Since  ∫ m_( , ')  (ρ_0⊗ρ_0)(, ') and ρ_0 are uniformly bounded away from 0 and ∞, it is enough to upper bound the 𝒞^0, α(Ω_0^δ)-norm of  the function 

    ↦∫ m_( , ) ρ_0().
  
Set 
, '∈Ω_0^δ and assume w.l.o.g. that 
-'^α≤σ_m(g)/4C_1,
 where  C_1>0 is the constant  such that 

    |_∇^2 g^*( a) ^2- _∇^2 g^*( b) ^2 |≤ C_1  ^2  a- b^α,  for all  a,  b,  ∈Ω_0

which due to the fact that ∇^2 g(') is Hölder  continuous.
Since C_d^-2/d+2 ( ρ_0()ρ_1(∇ g^*()) )^-1/d+2  is also a Hölder  continuous function, here exists C_2>0 

    |C_(')-C_()| ≤ C_2-'^α^2/d+2.

We aim at showing that 

    ∫ m_( , ) ρ_0()≤∫ m_(' , ) ρ_0() + C - '^α,

for some constant C>0. We call 

    𝒮_ := {∈Ω_0:  C_()+C_( z) -1/2-_∇^2 g^*() ^2- 1/2-_∇^2 g^*() ^2>0}

and observe that, due to (<ref>), there exists C_3>0 such that  sup_ u∈𝒮_ u- z≤ C_3 ^1/d+2.
Hence, we can find _δ>0 such that  ⋃_ u∈Ω^δ𝒮_ u⊂Ω^δ/2 for all ≤_δ. The rest of the proof always assume that ≤_δ.  Hence,

    ∫ m_( , ) ρ_0()
       = 1/2 ∫_Ω_0(C_()+C_( z) -1/2-_∇^2 g^*() ^2- 1/2-_∇^2 g^*() ^2 )_+  ρ_0()ℓ_d()
        = 1/2 ∫_^d(C_()+C_( z) -1/2-_∇^2 g^*() ^2- 1/2-_∇^2 g^*() ^2 )_+  ρ_0()ℓ_d().

 Making the change of variables '  = T_,'()=  +'-, 
one gets 

    ∫ m_( , ) ρ_0()
        
        = 1/2 ∫_^d(C_(' +  - ' )+C_( z) -1/2'-'_∇^2 g^*(' +  - ' ) ^2- 1/2'-'_∇^2 g^*() ^2 )_+ 
               ρ_0('-'+) ℓ_d(').


Via (<ref>) and (<ref>), the bound

    ∫ m_( , ) ρ_0()
       ≤1/2 ∫_^d(C_(')+C_( z') -1/2'-'_∇^2 g^*(' ) ^2-1/2'-'_∇^2 g^*(') ^2
                + 2  C_2^2/d+2-'^α+ 2 C_1'-'^2-'^α)_+
    ρ_0('-'+) ℓ_d(')

holds. 
Let 𝒮' be the set where the quantity inside the integral is positive and 

    C_5=(4 C_1/σ_m(g)(  C'σ_m(g)/4C_1+ 1/C_d^1/d+2λ^2/d+2))^1/2.
 Since -'^α≤σ_m(g)/4C_1, the nesting 
𝒮' ⊂𝔹( z',  ^1/d+2C_5)
is derived.  This combined with the bound on the regularity of the density
ρ_0('-'+)≤ρ_0(')+C_4-'^α
yield 

    ∫ m_( , ) ρ_0() 
       ≤1/2 ∫_^d(C_(')+C_( z') -1/2'-'_∇^2 g^*(' ) ^2-1/2'-'_∇^2 g^*(') ^2)_+ ρ_0(') ℓ_d(')
       +C_6 -'^α,

where 

    C_6 =2C_2C_5^dℓ_d(𝔹(0,1)) Λ +2C_1 C_5^d+2Λ +C_5 sup_'∈Ω^δ∫ m_(, ') ℓ_d(),

 which is finite (see the proof of  <ref>). 










Using (<ref>) we get 

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_()+C_( z)+ C_1/2-^2 '-^α -1/2_∇^2 g() ^2- 1/2_∇^2 g(') ^2 )_+  dρ_0() .

Using 

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_()+C_( z')+ C'^2/d+2'^α
                + C_1/2^2 '^α -1/2_∇^2 g() ^2- 1/2_∇^2 g(') ^2 )_+  dρ_0() .



Since '^α≤σ_m(g)/2C σ_M(g), it holds that ℓ_d(𝒮)≤ C ^d/d+2 and max_∈𝒮≤ C' ^1/d+2. As a consequence, 

    ∫ m_( , ) ρ_0()
           ≤1/2 ∫(C_()+C_( z')  -1/2_∇^2 g() ^2- 1/2_∇^2 g( z') ^2 )_+  dρ_0()
            + '^α(C_1 C^2σ_M(g)/2 + C' C”)

Change variables =̆T()=+'  and assume that  is small enught so that T(𝒮)∩^d∖Ω_0=∅. Then 

    1/2 ∫(C_()+C_( z')  -1/2_∇^2 g() ^2- 1/2_∇^2 g( z') ^2 )_+ dρ_0()
       =1/2 ∫(C_(-̆')+C_(') -1/2-̆'_∇^2 g(-̆') ^2- 1/2-̆'_∇^2 g( 0) ^2 )_+  ρ_0(-̆') ℓ_d()̆.

Again, using (<ref>) and (<ref>) and 
repeating the previous argument, 
we get 

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_(-̆')+C_(') -1/2-̆'_∇^2 g(-̆') ^2- 1/2-̆'_∇^2 g( 0) ^2 )_+  ρ_0(-̆') ℓ_d()̆
            + C'^α,

for some C>0. 
Recalling that ρ_0 is Hölder continuous, we also have |ρ_0(-̆')-ρ_0()̆|≤ C'^α, so that

    ∫ m_( , ) ρ_0()
        
       ≤1/2 ∫(C_(-̆')+C_(') -1/2-̆'_∇^2 g(-̆') ^2- 1/2-̆'_∇^2 g( 0) ^2 )_+  ρ_0()̆ℓ_d()̆ + C'^α
       = ∫ m_(' , ) ρ_0()+ C'^α.

The last step above wasn't clear to me; details would be enjoyable!


This yields the first part of the Hölder inequality statement. 
The reverse inequality holds due to symmetry,  exchanging  by '. 


Proof of <ref> 
By (<ref>), Lemma <ref> and the fact that the support of m_ε decreases with rate ^2/d+2, we have 



    m_ε ( x,  x')=1/( C_() -1/2-'_∇^2 g^*() ^2 +o(^2/d+2) )_+ .


Set now δ>0 and ∈Ω_0^δ. Then there exists _0>0 small enough such that 

    ∫_Ω_0^δ( C_()-1/2-'_∇^2 g^*() ^2)_+ ℓ_d(')
        = ∫( C_()-1/2-'_∇^2 g^*() ^2 )_+ ℓ_d('),

for all ≤_0. Now we finish the argument. The change of variables 
=̆  [D^2g^* ()]^1/2 (-')
 yields

    ∫ m_(, ') dρ(')   = ρ_0() (1+o(1))/(D^2g^* ())^1/2∫_Ω_0^δ( C_()-^2/2)_+ dℓ_d()̆
       = ρ_0() (1+o(1))/(D^2g^* ())^1/2 (ρ_0() ρ_1[∇ g^*()]))^1/2.

The relation (D^2g^* ())= ρ_0() /ρ_1[∇ g^*()] yields
∫ m_ (·, ') ρ_0(') -1_𝒞(Ω_0^δ)→ 0,
which in turn implies
ρ_ε -1_𝒞(Ω_0^δ)→ 0. 


<cit.> established the existence and uniqueness (up to additive constants) of a convex function ϕ_ε, for every ε, such that its gradient ∇ϕ_ε pushes forward the measure ρ_ερ_0 to ρ_0 That is, 
∫  f(∇ϕ_)  ρ_ρ_0= ∫ f  ρ_0, for all f∈𝒞(Ω_0).
In particular ϕ_ solves the boundary value problem 

    (∇^2 ϕ_)=ρ_0 ρ_/ρ_0(∇ϕ_)    in Ω_0,   ∇ϕ_ (Ω_0)= Ω_0

and its convex conjugate ϕ_^* solves 

    (∇^2 ϕ_^*)=ρ_0 /ρ_0(∇ϕ_^*) ρ_(∇ϕ_^*)   in Ω_0,   ∇ϕ_^* (Ω_0)= Ω_0.

Combining Theorem <ref> with Lemma <ref>, we can ensure the existence of _δ>0 such that

    ∇ϕ__𝒞^1, α(Ω_0^δ),  ∇ϕ_^*_𝒞^1, α(Ω_0^δ)≤ C(δ),   for all ≤_δ.

Therefore, standard stability theory (see e.g., <cit.>) and the Arzelà–Ascoli theorem yield

    ∇ϕ_^*- Id_𝒞^1(Ω_0^δ),  ∇ϕ_- Id_𝒞^1(Ω_0^δ)→ 0,   as → 0.

This limit is crucial to show the following result. 


    Let Assumption <ref> holds. There exists δ_0 such that for all  0<δ≤δ_0 there exists a function ω=ω_δ, such that ω()→ 0 as → 0, and C=C(δ)>0 such that
    
    |  D(∇ϕ_^*(),∇ g^*(∇ϕ_^*(')))- -'_∇^2 g^*(')^2 |≤   C -'^2( -'^α+ω()),

    and 
    
    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- -'_∇^2 g^*(')^2 |≤   C -'^2( -'^α+ω()),

    for all , '∈Ω^δ_0={∈Ω_0:   dist  (, ∂Ω_0)≥δ}.


We call ω()= ∇ϕ_^*- Id_𝒞^1(Ω_0^δ)^α/2, which, by (<ref>), tends to 0 as → 0. 
    Lemma <ref> first and then equation (<ref>) imply that 

    D(∇ϕ_^*(),∇ g^*(∇ϕ_^*(')))
       = 1/2∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2 +𝒪( ∇ϕ_^*() -∇ϕ_^*('))^2+α)
       = 1/2∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2 +𝒪( -'^2+α)

for all , '∈Ω_0^δ. Hence, the proof will be complete after showing that 

    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2--'_∇^2 g^*(')^2|≤ C ω() -'^2 .

for all , '∈Ω_0^δ and some C>0. 
First we exchange ∇^2 g^*(∇ϕ_^*(')) by ∇^2 g^*('). 
Since ∇^2 g^* is Hölder continuous,  we obtain 

    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- ∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(')^2|≤ω() -'^2.

Then we exchange ∇ϕ_^* by the identity 

    ∇ϕ_^*()    -∇ϕ_^*(')_∇^2 g^*(')^2 
       =  [∇ϕ_^*- Id]() -[∇ϕ_^*- Id](')_∇^2 g^*(')^2+ -'_∇^2 g^*(')^2
            + 2⟨ [∇ϕ_^*- Id]() -[∇ϕ_^*- Id]('),   -'⟩_∇^2 g^*(')
       ≤-'_∇^2 g^*(')^2(1+3ω()+2ω() ),

which implies (<ref>)  and concludes the proof. 




The following result finds a measure μ_ with the correct marginals and provides the exact description of its density. 


 Let Assumption <ref> hold and   assume further that Ω_0 is convex. Define 
    μ_:= (∇ϕ_×∇ϕ_)_#(ξ_ (ρ_0 ⊗ρ_0))

 and
 
    u_(, '):= μ_/ (ρ_0⊗ρ_0)(, ')=    ξ_(∇ϕ_^*(), ∇ϕ_^*('))  /ρ_(∇ϕ_^*())ρ_(∇ϕ_^*(')).

 Then μ_ε belongs to Π(ρ_0, ρ_0) and, as a consequence, the measure 
π_^(1)=( Id×∇ g )_#μ_,
belongs to Π(ρ_0, ρ_1) and it
has density 

    π_^(1)/ (ℓ_d⊗ℓ_d)(, )= u_(,∇ g ()) ρ_0() ρ_1().



    We can find the shape of its density by computing the following integral for an arbitrary bounded continuous function f:

    ∫ f(∇ϕ_(), ∇ϕ_('))  ξ_(, ')   (ρ_0 ⊗ρ_0)(, ')
       = ∫
            f(, ' )  ξ_(∇ϕ_^*(), ∇ϕ_^*('))  ρ_0(∇ϕ_^*()) ρ_0(∇ϕ_^*(') )
         /[(∇^2ϕ_^*())(∇^2ϕ_^*('))]^-1ℓ_d() ℓ_d(')
       =∫f(, ')  ξ_(∇ϕ_^*(), ∇ϕ_^*('))  ρ_0(∇ϕ_^*()) ρ_0(∇ϕ_^*(') )/ρ_0(∇ϕ_^*()) ρ_0(∇ϕ_^*(')) ρ_(∇ϕ_^*())ρ_(∇ϕ_^*('))ρ_0() ρ_0(')
    
       =∫f(, ')  ξ_(∇ϕ_^*(), ∇ϕ_^*('))  /ρ_(∇ϕ_^*())ρ_(∇ϕ_^*('))ρ_0() ρ_0(').

Since the latter holds for any continuous function f,  (<ref>) holds. 

It is easy to check that by construction 

    ∫ f() u_(, ') ρ_0() ρ_0(')   =∫ f(∇ϕ_())  ξ_(, ')   (ρ_0 ⊗ρ_0)(, ')
       =∫ f(∇ϕ_())    ρ_()
       =∫ f()    ρ_0(),

so that u_(, ')  (ρ_0 ⊗ρ_0)(, ') belongs to Π(ρ_0, ρ_0). 

Now we prove some important properties of u_.



    Let Assumption <ref> hold and   assume further that Ω_0 is convex. 

Then there exists δ_0>0 such that 
for every δ∈ (0, δ_0) there exists _δ such that 

    ∫_Ω^δ_0  u_(, · ) dρ_0()_𝒞(Ω^δ,c_0)≤2/3

for all ≤_δ. 


Due to (<ref>), there exists δ_0 such that for all δ≤δ_0 there exists  _δ such that 

    {'∈Ω_0: ∃ ∈Ω_0^δ/2:   m_(, ')>0 }⊂Ω_0^δ/4

for all ≤_δ. Equation (<ref>) implies that, for a maybe smaller _δ, 

    ∇ϕ_^*( Ω_0^δ)⊂Ω_0^δ/2,     ∇ϕ_( Ω_0^δ/4)⊂Ω_0^δ/8   and  ∇ϕ_^*( Ω_0^δ/8)⊂Ω_0^δ/16

for all ≤_δ. As a consequence, 

    𝕎_,δ   :={'∈Ω_0: ∃ ∈Ω_0^δ :   m_(∇ϕ_^*(), ')>0 }
        =   {'∈Ω_0: ∃ ∈∇ϕ_^*(Ω_0^δ):   m_(, ')>0 }
       ⊂{'∈Ω_0: ∃ ∈Ω_0^δ/2:   m_(, ')>0 }
       ⊂Ω_0^δ/4

and 

    {'∈Ω_0: ∃ ∈Ω_0^δ :   m_(∇ϕ_^*(), ∇ϕ^*('))>0 }=∇ϕ (  𝕎_,δ  ) ⊂∇ϕ ( Ω_0^δ/4  ) ⊂Ω_0^δ/8.

As a consequence, 
∫_Ω^δ_0  u_(, ' ) dρ_0()=0 for every ' ∈Ω_0^δ/8,c and ≤_δ. Therefore, we can assume that ' ∈Ω_0^δ/8, ∈Ω_0^δ and ∇ϕ_^*(') , ∇ϕ_^*() ∈Ω_0^δ/16. Due to Lemma <ref>, <ref>,

    u_(, ')= ξ_(∇ϕ_^*(), ∇ϕ_^*('))  / 1+o(1) = m_(∇ϕ_^*(), ∇ϕ_^*('))  / 1+o(1) ,

where the error term is uniform for , '∈Ω_0^δ/8. On the other hand,

    | C_(∇ϕ_ε^*())-C_()|≤^2/d+2∇ϕ_^*- Id_𝒞^1(Ω_0^δ)^α =o( ^2/d+2),

where the last term is a consequence of (<ref>).  We observe that for , '∈Ω_0^δ/8 such that the inequality 
m_(∇ϕ_^*(), ∇ϕ_^*('))>0 holds,
equation (<ref>) implies
∇ϕ_^*()-∇ϕ_^*(')^2 ≤ C ^2/d+2,
and, by (<ref>), also 

    -'^2 ≤ C ^2/d+2.

 Hence, (<ref>) yields 

    -'^2_∇^2 g^*()= -'^2_∇^2 g^*(')  + o( ^2/d+2) ,

where the error term is uniform for , '∈Ω_0^δ/8 such that  m_(∇ϕ_^*(), ∇ϕ_^*('))>0. Hence, the previous display, (<ref>) and (<ref>) imply 

    m_(∇ϕ_^*(), ∇ϕ_^*(')) = 1/( C_(')-1/2-'^2_∇^2 g^*(')+ o( ^2/d+2))_+

for all , '∈Ω_0^δ/8 such that  m_(∇ϕ_^*(), ∇ϕ_^*('))>0.
This,  combined with (<ref>), yields

    ∫_Ω_0^δ  u_(, ') ρ_0()
       = ∫_Ω_0^δ  m_(∇ϕ_^*(), ∇ϕ_^*(')) ρ_0()  / 1+o(1) 
       ≤1/∫_Ω_0^δ( C_(')-1/2-'^2_∇^2 g^*(')+ o( ^2/d+2))_+ ρ_0()  / 1+o(1) 
    
       ≤1/  (ρ_0(')+o(1))∫_Ω_0^δ( C_(')-1/2-'^2_∇^2 g^*(')+ o( ^2/d+2))_+ ℓ_d()  / 1+o(1) .

Changing variables =̆ T()= [∇^2 g^*(')]^1/2 (-') to get (up to uniform o(1) terms) 

    ∫_Ω_0^δ   u_(, ') ρ_0() ≤1/ε·ρ_0(')/(∇^2 g^*('))^1/2∫_T(Ω_0^δ)( C_(')-^2/2)_+ ℓ_d().

Without losing generality we can assume that '= 0. Then 0∉ T(Ω_0^δ). Since T(Ω_0^δ) is convex, there exist a vector $̌ and a  hyperplaneH={ : ⟨, ⟩̌=0 } such thatT(Ω_0^δ)⊂H_+:={ : ⟨, ⟩̌≤0 } . As a consequence, up too(1)additive terms,

    ∫_Ω_0^δ   u_(, ') ρ_0()    ≤1/ε·ρ_0(')/(∇^2 g^*('))^1/2∫_H_+( C_(')-^2/2)_+ ℓ_d()
       = 1/ε·ρ_0(')/2 (∇^2 g^*('))^1/2∫( C_(')-^2/2)_+ ℓ_d()
    (by Proposition <ref>)    =1/ε·ρ_0(')/2 (∇^2 g^*('))^1/2|C_(')|^d+2/2 C_d 
       =1/ε·ρ_0(')/2 (∇^2 g^*('))^1/2( ^2/d+2/C_d^2/d+2(ρ_0(')ρ_1[∇ g^*(')])^1/d+2)^d+2/2 C_d
       =ρ_0(')/2 (∇^2 g^*('))^1/21/(ρ_0(')ρ_1[∇ g^*(')])^1/2.

Since,(∇^2 g^*)=ρ_0/ρ_1(∇g^*) , we get∫_Ω_0^δ   u_(, ') ρ_0()≤1/2+o(1), and the claim follows. 



 §.§ Valid coupling for the frame Ω_0^δ,c of a convex Ω_0

Forδ>0, we recall the definition (<ref>) and consider the truncated function

    u_χ_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0),

in an effort to avoid the problematic regions ofΩ_0×Ω_0where the candidate for coupling is not regular enough. Notice how we allow one of the variables,for example, to be close to the boundary as long as the other one,', is not. In principle, this may seem inadvisable; however, there is no issue since the support of the coupling will concentrate around the set='. Refer to the proof of Lemma <ref>  above or that of Lemma <ref> below for further details. Therefore,  the only truly problematic set is(Ω^δ, c_0×Ω^δ, c_0).

Thus, in seeking a true coupling in(Ω_0×Ω_0)we search for a densityv_such  that1=∫  v_(, ') ρ_0()= ∫  v_(, ') ρ_0('),   for all , '∈Ω_0,and we want it of the formv_= u_χ_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)+  h_ε, δχ_(Ω^δ,c_0×Ω^δ,c_0 ),for some positive functionh_, δ. Irrespective of h_ε, δ, for'∈Ω^δ_0the “coupling” condition1= ∫ v_(,') dρ_0()must hold. Hence, we need only to fith_ε, δsuch that 
that1= ∫_Ω^δ,c_0   h_ε, δ(, ') dρ_0() + ∫_Ω^δ_0  u_(, ') dρ_0(),   for all '∈Ω^δ,c_0and 

    1= ∫_Ω^δ,c_0   h_ε, δ(, ') dρ_0(') + ∫_Ω^δ_0  u_(, ') dρ_0('),   for all ∈Ω^δ,c_0.

We find firsth_, δsuch that condition (<ref>) holds and then we construct a coupling based on this. In order to simplify the notation we defineq_δ():=1-(∫_Ω^δ_0  u_(, ') dρ_0('))χ_Ω^δ,c_0() ∈ (0,1].Note that due to Lemma <ref>, we can assume thatεis sufficiently small so that

    1/3≤ q_δ(𝐱) ≤ 1,   for all ∈Ω^δ, c.

Next, we show that (<ref>) holds forΩ^δ, c.

    Let Assumption <ref> hold and Ω_0 be convex.  there exist δ_0>0 and θ∈ (0,π) such that for all δ≤δ_0 and ∈Ω^δ,c there exists a cone
    𝒞_,  v,r, θ
    with vertex ,  height r=r_δ>0 and angle θ such that 𝒞_,  v,r, σ⊂Ω^δ,c.  

    Let δ_0 be such that Ω_0^δ exists and is convex and nonempty,  δ≤δ_0 and  r'_δ= δ/4. Then, for ∈Ω^δ,c_0 and r' ≤ r_δ', the ball 𝔹(𝐱, r') intersects at most one of ∂Ω_0^δ or ∂Ω_0, but not both simultaneously. In the last case, the claim holds by Assumption <ref>. Therefore, we can assume that 𝔹(𝐱, r') ∩∂Ω_0=∅. 
 Since Ω_0^δ is convex and ∉Ω_0^δ, there exists a separating half-space H_+={: ⟨,  v⟩≥ a} such that ∈ H_+ and Ω_0^δ∩  H_+=∅. Therefore, 
 𝔹(𝐱, r'  ) ∩ H_+⊂𝔹(𝐱, r  ) ∩Ω^δ,c_0,
for every r'≤ r_δ. Setting θ=π/4 and 
 r=max{ x_1:  (x_1, …, x_d)≤ r_δ'    and  (x_1, …, x_d)cos(1/2θ) ≤ 
x_1 }>0
the result follows. 


  

Due to Lemma <ref> and (<ref>), the conditions of Lemma <ref> hold fors=q_δandΩ=Ω^δ,c. As a consequence, forsmall enough there existsc_,δ:Ω^δ,c →such thatq_δ()=1/∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ρ_0(') ,   for all ∈Ω^δ,c_0.Moreover,
there existsC=C(δ_0)such that for everyδ≤δ_0there exists_δ>0such that for≤_δ, the functionc_, δexists,  

     
    ε^2/d+2 C^-1≤  c_, δ() ≤ε^2/d+2 C,    ε^d/d+2 C^-1≤ℓ_d(𝒮_ε,δ, )≤ Cε^d/d+2, 
       and
       1/∫_Ω^δ,c_0 (c_, δ ()-D(, ∇ g^*(') )_+ρ_0() ≥ C^-1,

     for every'∈Ω_0^δ,cwhere𝒮_ε,δ, = { '∈Ω_0:  c_, δ ()-D(, ∇g^*(')  ≥0}.To simplify the exposition, we first introduce a few recurrent quantities. Define

    M_ε, δ (𝐚, 𝐛) := (c_ε, δ(𝐚) - D(𝐚, ∇ g^*(𝐛))_+/ε,

and the density 

    h_ε, δ(𝐱, 𝐱') := ∫_Ω_0^δ,cM_ε, δ(𝐱', 𝐳) M_ε, δ(𝐱, 𝐳)/∫_Ω_0^δ,c M_ε, δ(𝐯', 𝐳)   dρ_0(𝐯')  dρ_0(𝐳),

with the associated measure given by

    dπ_ε^(2)(𝐱, 𝐱') = h_ε, δ(𝐱, 𝐱')   dρ_0(𝐱)   dρ_0(𝐱').

 Let Assumption <ref> hold. Then  
     q_δ()= ∫_Ω^δ,c_0  h_ε, δ(, ')ρ_0(')    and    q_δ(')= ∫_Ω^δ,c_0   h_ε, δ(, ')ρ_0(). 
    As a consequence, 
    v_ (ρ_0 ⊗ρ_0)∈Π(ρ_0, ρ_0),
    where
    
    v_= [u_χ_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)+ h_, δχ_(Ω^δ,c_0×Ω^δ,c_0 ) ]
    
    Set 
    and compute 
    
    ∫_Ω^δ,c_0  h_ε, δ(, ')ρ_0(')   = ∫_Ω^δ,c_0∫_Ω^δ,c_0 M_, δ (', )  M_, δ (, )  /∫_Ω^δ,c_0 M_, δ('̌, ) ρ_0('̌) ρ_0()  ρ_0(')
       =   ∫_Ω^δ,c_0∫_Ω^δ,c_0M_, δ (', )  ρ_0(') M_, δ (, )  /∫_Ω^δ,c_0 M_, δ('̌, ) ρ_0('̌) ρ_0()  
       = ∫_Ω^δ,c_0 M_, δ (, )  ρ_0() 
       =q_δ().

The same holds for the other variable by symmetry. 

We give some estimates on the function h_, δ(, '), which will be useful for the sequel.  

    Let assumption <ref> hold. Then there there exists  δ_0>0 and C>0 such that for every δ∈ (0, δ_0)  there exist _δ
such that  

    ∫_Ω^δ,c_0×Ω^δ,c_0 D(,∇ g^*(')) h_, δ(, ')   (ρ_0⊗ρ_0)(, ') ≤ C  ℓ_d(Ω_0^δ,c ) ^2/d+2 ,

and 

    /2∫_Ω^δ,c_0×Ω^δ,c_0   (h_(, '))^2 (ρ_0⊗ρ_0)(, ')  ≤ C ℓ_d(Ω_0^δ,c ) ^2/d+2 ,

for all ≤_δ.
 

By (<ref>),  it is sufficient to verify (<ref>) and (<ref>) by substituting  h_, δ by  M_, δ. 



First we give some estimates on the support of M_,δ. The set of (, ') such that  ∫_Ω^δ,c_0 M_, δ (', )  M_, δ (, )  ρ_0()>0 is called 
A_δ, . It can be also described as the set of (, ') such that  there exists ∈Ω_0^δ,c belonging to supp(M_, δ (, · ))∩ supp(M_, δ (', · )), i.e.,  satisfying
M_, δ (', )  M_, δ (, )>0.
Recall that 
∫_Ω^δ,c_0 M_, δ (', )  M_, δ (, )  ρ_0()>0. 
(<ref>) implies that there exists C>0 such that 
supp(M_, δ (, · ))⊂𝔹(, C^2/d+2),
so that 
A_δ,⊂{ (, ')∈ (Ω^δ,c_0)^2:  𝔹(, C^2/d+2)∩𝔹(', C^2/d+2)≠∅}.
Therefore, by increasing C, we have 

    A_δ,⊂{ (, ')∈ (Ω^δ,c_0)^2:  '∈𝔹(, C^2/d+2)}.

Now we give some estimates on the maximum value that h_, δ can take. 
According to  (<ref>),  there exists δ_0>0 and C(δ_0)>0 such that for every δ∈ (0, δ_0)  there exist _δ
such that for every ≤_δ, it holds that 
M_, δ (', )  ≤   C(δ_0)^-d/d+2. 
Hence,  

    ∫_Ω^δ,c_0 M_, δ (', ) M_, δ (, ) dρ_0()   ≤ C(δ_0)^-d/d+2∫_Ω^δ,c_0 M_, δ (, ) ρ_0()
        = C(δ_0)^-d/d+2 q_δ()
       ≤  C(δ_0)^-d/d+2,

which implies 

    max_h_, δ(', )≤ C(δ_0) ^-d/d+2.

At this point, we have all the ingredients to prove (<ref>) and (<ref>). Firstly,  
we find an upper bound for (<ref>) as follows: 

    /2∫_Ω^δ,c_0×Ω^δ,c_0     (h_(, '))^2 (ρ_0⊗ρ_0)(, ') 
       ≤/2∫_Ω^δ,c_0max_h_, δ(', )∫_Ω^δ,c_0  h_, δ(, ')ρ_0() ρ_0(')  
       = /2∫_Ω^δ,c_0max_h_, δ(, ) q_δ()ρ_0(') 
       ≤/2∫_Ω^δ,c_0max_h_, δ(', )ρ_0(') 
    (by (<ref>))    ≤C(δ_0) ^2/d+2/2Λℓ_d(Ω^δ,c_0).

Hence, (<ref>) holds. To prove (<ref>), we use first Lemma <ref> and then (<ref>)  to obtain 

    ∫_Ω^δ,c_0×Ω^δ,c_0    D(,∇ g^*(')) h_, δ(, ')  (ρ_0⊗ρ_0)(, ') 
       ≤σ_M(g)/2∫_Ω^δ,c_0×Ω^δ,c_0-'^2 h_, δ(, ')   (ρ_0⊗ρ_0)(, ')
       ≤ C(δ_0, g) ^2/d+2∫_Ω^δ,c_0×Ω^δ,c_0  h_, δ(, ')   (ρ_0⊗ρ_0)(, ').

Since, via Lemma <ref>, we have
∫_Ω^δ,c_0 h_, δ(, ')  ρ_0() =q_δ(') ≤ 1,
wich yields

    ∫_Ω^δ,c_0×Ω^δ,c_0    D(,∇ g^*(')) h_, δ(, ')  (ρ_0⊗ρ_0)(, ') ≤ C(δ_0, g) ^2/d+2Λℓ_d(Ω^δ,c_0 ).

The result follows. 


 §.§ Limit for convex supports


Let us define the measure ν_ε(,') = v_(, ')   (ρ_0⊗ρ_0)(, '),
meaning that if we defineπ_ε:=(Id ×∇g^*)_# ν_then we readily obtain

    π_(x,y) =(Id ×∇ g^*)_#ν_=v_(, ∇ g())   (ρ_0⊗ρ_1)(, ).


In this section, we find the limit of ^-2/d+2 (ℍ_( π_)-𝒲_2^2(ρ_0, ρ_1)), whereℍ_ ( π)= ∫-^2 π(,)   (ρ_0⊗ρ_1)(,) +  π/ (ρ_0⊗ρ_1)_L^2(ρ_0⊗ρ_1)^2,in the case whereΩ_0is a convex domain, which will be provided by the following asymptotic bound.  

    Let Assumption <ref> hold and Ω_0 be convex. Then 
    lim sup_→ 0^+𝒯_2, ε, (·)^2(ρ_0, ρ_1) - 𝒲_2^2(ρ_0, ρ_1)/^2/d+2≤d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0(). 
First, as in the proof of Lemma <ref>, there exists δ_0 such that for all δ≤δ_0 there exists  _δ such that  

    {'∈Ω_0: ∃ ∈Ω_0^δ :   m_(∇ϕ_^*(), ∇ϕ^*('))>0 }⊂Ω_0^δ/8, 
       ∇ϕ_^*( Ω_0^δ)⊂Ω_0^δ/2,     ∇ϕ_^*( Ω_0^δ/4)⊂Ω_0^δ/8   and  ∇ϕ_^*( Ω_0^δ/8)⊂Ω_0^δ/16

for all ≤_δ. Therefore, from now on we assume that ≤_δ for 0<δ≤δ_0. 
We analyze first the linear term, where π_0 is the minimizer of (<ref>). By duality (<ref>), it holds that 

    1/2∫-^2  (π_-π_0)(,)
    =  ∫1/2-^2 π_(,) - ∫(1/2^2-g^*()) ρ_0()-∫(1/2^2-g())  ρ_1().

Since π_∈Π(ρ_0, ρ_1), we get 

    1/2∫-^2  (π_-π_0)(,)   =∫ D(, ) π_(,)
        
       =∫ D(, ∇ g^*(')) ν_ (,').

We observe that 

    ∫ D(,∇ g^*(')) ν_(,') 
       = ∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) D(,∇ g^*(')) u_(, ')   (ρ_0⊗ρ_0)(, ')_𝕃_1
            + ∫_(Ω^δ,c_0×Ω^δ,c_0) D(,∇ g^*(')) h_, δ(, ')   (ρ_0⊗ρ_0)(, ')_𝕃_2 .

We denote the first term by 𝕃_1 and the second by 𝕃_2. To bound 𝕃_2, Lemma <ref> ensures that there exist δ_0 > 0 and C > 0 such that for every δ∈ (0, δ_0), there is an ε_δ satisfying   

    𝕃_2=∫_(Ω^δ,c_0×Ω^δ,c_0) D(,∇ g^*(')) h_, δ(, ')   (ρ_0⊗ρ_0)(, ')≤ Cρ_0(Ω_0^δ) ^2/d+2

for every ≤_δ.
For 𝕃_1 we observe that Lemma <ref> (point  <ref>), (<ref>) and (<ref>), imply  that
𝕃_1≤1/1+o(1)∫_(Ω^δ'_0×Ω^δ'_0) D(∇ϕ_^*(),∇ g^*(∇ϕ_^*('))) m_(, ')   (ρ_0⊗ρ_0)(, '),  
for some 0<δ'< δ. 
By 
Lemma <ref>, we have 
|  D(∇ϕ_^*(),∇ g^*(∇ϕ_^*(')))- 1/2-'_∇^2 g^*(')^2 |≤   C -'^2( -'^α+ω()),  
for all , '∈Ω_0^δ',
where ω()→ 0 as → 0.



 
  

As a consequence, up to o(1) terms, we have the bound

    𝕃_1   ≤∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 m_(, ')   (ρ_0⊗ρ_0)(, ')
            + C ∫1/2-'^2 ( -'^α+ω())  m_(, ')   (ρ_0⊗ρ_0)(, '),

where, due to (<ref>), the last term is of order  o(ε^2/d+2). Therefore, it holds that

    𝕃_1  ≤∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 m_(, ')   (ρ_0⊗ρ_0)(, ') + o(ε^2/d+2).






 Lemma <ref> yields the estimate
 
    m_(, ')=1/( C_(') -1/2-'_∇^2 g^*(')^2)_+ + ω() ^-d/d+2,

which implies

    𝕃_1 
        ≤^-1∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 ( C_(') -1/2-'_∇^2 g^*(')^2)_+   (ρ_0⊗ρ_0)(, ')
    + ω() ^-d/d+2∫_ supp(m_)1/2-'_∇^2 g^*(')^2   (ρ_0⊗ρ_0)(, ')
        +o(^2/d+2).

The bound (<ref>) implies 

    𝕃_1 
        ≤^-1∫_(Ω^δ'_0)^21/2-'_∇^2 g^*(')^2 ( C_(') -1/2-'_∇^2 g^*(')^2)_+   (ρ_0⊗ρ_0)(, ')
    
        +o(^2/d+2).

By adding and subtracting C_(') inside the integral, the bound

    𝕃_1    ≤    -^-1∫_(Ω^δ'_0)^2( C_(') -1/2-'_∇^2 g^*(')^2)_+^2  (ρ_0⊗ρ_0)(, ')
           
         + ^-1∫_(Ω^δ'_0)^2 C_(') ( C_(') -1/2-'_∇^2 g^*(')^2)_+  (ρ_0⊗ρ_0)(, ')_𝔻 +o( ^2/d+2)

follows. 
By using   (<ref>) and Lemma <ref> (point  <ref>) again we get 

    𝔻≤∫ C_(') u_(, ')  (ρ_0⊗ρ_0)(, ')
    
        + w() ^-d/d+2∫_C(')≥1/2-'^2 C_(')  (ρ_0⊗ρ_0)(, ') + o( ^2/d+2),

which yields 
𝔻≤∫ C_(') ρ_0(')+ o( ^2/d+2)
and  a fortiori
    𝕃_1 ≤    -^-1∫_(Ω^δ'_0)^2( C_(') -1/2-'_∇^2 g^*(')^2)_+^2  (ρ_0⊗ρ_0)(, ')
    
        + ∫ C_(') ρ_0(')  +o( ^2/d+2)



Now we deal with the penalty term. As before, we divide 
 π/ (ρ_0⊗ρ_1)_L^2(ρ_0⊗ρ_1)^2=∫ (v_(, '))^2 (ρ_0⊗ρ_0)(, ') 

into two additive terms
ℚ_1= ∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)  (u_(, '))^2 (ρ_0⊗ρ_0)(, ') 
and  

    ℚ_2= ∫_Ω^δ,c_0×Ω^δ,c_0  (h_, δ(, '))^2 (ρ_0⊗ρ_0)(, ')≤   C^2/d+2ℓ_d(Ω^δ,c_0),

where the bound is derived from Lemma <ref>, and the constant C is independent of δ.
 We bound now ℚ_1.  
 Since C_() is Hölder continuous with constant C^2/d+2 and owing to (<ref>), it holds that 
 
    C_(∇ϕ_^*()) - C_()≤ C ^2/d+2ω() ,    with ω()→ 0.

  By 
Lemma <ref>, we also have 
|∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- -'_∇^2 g^*(')^2 |≤ C -'^2( -'^α+ω()) 
and 
|∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*())^2- -'_∇^2 g^*()^2 |≤ C -'^2( -'^α+ω()). 
 Since ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0 implies
 
    C^1/d+2   ≥∇ϕ_^*()-∇ϕ_^*(')
       ≥-'-[∇ϕ_^*- Id]()-[∇ϕ_^*- Id](')
       ≥-'(1-∇ϕ_^*- Id_𝒞^1(Ω_0^δ)),


 were we used the lower bound on ∇^2 g^*() and the fact that all norms on ℝ^d are equivalent (which is hidden in the constant).
 Now, 
 using (<ref>) we get
 
    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*('))^2- -'_∇^2 g^*(')^2 |≤ C ω() ^2/d+2

and 

    |∇ϕ_^*() -∇ϕ_^*(')_∇^2 g^*(∇ϕ_^*())^2- -'_∇^2 g^*()^2 |≤  Cω() ^2/d+2,

for all (, ')∈ (Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) such that ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0 and
a function ω=ω_δ such that ω()→ 0 as → 0.  Since ∇^2 g^* is uniformly continuous, 
by the same arguments as before, 
|-'_∇^2 g^*(')^2 - -'_∇^2 g^*()^2 |≤ω() ^2/d+2,  
for all (, ')∈ (Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) such that ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0. Therefore, this last fact combined with (<ref>), (<ref>) and (<ref>) implies 
ξ_(∇ϕ_^*(), ∇ϕ_^*(')) 
    ≤1/(C_()-1/2-'_∇^2 g^*(')^2 
    +ω() ^2/d+2)_+.  

for some  function ω=ω_δ such that ω()→ 0 as → 0. 
This combined with the point <ref> of
Lemma <ref> and (<ref>) yields

    u_(, ')   =ξ_(∇ϕ_^*(), ∇ϕ_^*('))  /ρ_(∇ϕ_^*())ρ_(∇ϕ_^*('))
       ≤1/(1+o(1))(C_()-1/2-'_∇^2 g^*()^2 
        +ω() ^2/d+2)_+.  
       ≤1/(1+o(1))(C_()-1/2-'_∇^2 g^*()^2 
         )_+ +ω() ^-d/d+2,

where the the error terms are uniform on the set of (, ')∈ (Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0) such that 
ξ_(∇ϕ_^*(), ∇ϕ_^*('))>0
and a fixed 0<δ≤δ_0. 
As a consequence,  it holds — up to o(^2/d+2) additive terms, that

    ℚ_1   = ∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)  (u_(, '))^2 (ρ_0⊗ρ_0)(, ') 
       ≤∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2 (ρ_0⊗ρ_0)(, ')+ o(^2/d+2)
       ≤∫_(Ω^δ'_0)^2(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2 (ρ_0⊗ρ_0)(, ')+ o(^2/d+2),
 
for 0<δ'<δ. 
The last display and (<ref>) yield 

    2 𝕃_1 +ℚ_1 ≤    -^-1∫_(Ω^δ'_0)^2( C_(') -1/2-'_∇^2 g^*(')^2)_+^2  (ρ_0⊗ρ_0)(, ')
    +
    2∫ C_(') ρ_0(') +o( ^2/d+2),

while  (<ref>) and (<ref>) give 
2 𝕃_2+ℚ_2≤ C^2/d+2ℓ_d(Ω^δ,c_0).
From the relation 

    ℍ_ ( π_)- 𝒲_2^2(ρ_0, ρ_1)=2 𝕃_1+2 𝕃_1+ℚ_1+ℚ_2
 we get the estimate

    ^-2/d+2((ℍ_ ( π_)- 𝒲_2^2(ρ_0, ρ_1) )
    ≤ 2^-2/d+2∫  C_(') ρ_0(') - ^d+4/d+2∫_(Ω^δ_0)^2(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ')_from 2𝕃_1+ℚ_1
    
       + C ℓ_d(Ω^δ,c_0)_from 2𝕃_2+ℚ_2+o(1)

and we conclude as in the proof of the lower bound. 

We now use the bound

    ^d+4/d+2/2∫_(Ω^δ_0×Ω_0) ∪ ( Ω_0×Ω^δ_0)(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ')
    ≥^d+4/d+2/2∫_(Ω^δ_0×Ω_0)(C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ')

and observe that for fixed δ>0 and   small enough the latter is equal to
^d+4/d+2/2∫_Ω^δ_0×^d (C_()-1/2-'_∇^2 g^*()^2 
     )_+^2   (ρ_0⊗ρ_0)(, ').
Solving the integral as in (<ref>), we obtain 

    ^d+4/d+2∫_Ω^δ_0×^d (C_()-1/2-'_∇^2 g^*()^2 
         )_+^2   (ρ_0⊗ρ_0)(, ') 
    ≍ 2/d+4C_d^ -2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0().
As a consequence, 

    ^-2/d+2(ℍ_ ( π_)- 𝒲_2^2(ρ_0, ρ_1)≤4/d+4C_d^ -2/d+2∫_Ω_0^δ(ρ_0()ρ_1[∇ g^*()])^-1/(d+2)ρ_0()
    
       +^2/d+2+ C ℓ_d(Ω^δ,c_0)+o(1),


 §.§ General case

The aim of this section is to prove Theorem <ref> in the general case, namely, to eliminate the assumption of convex support. To this end, we will construct the aforementioned “stained glass” structure. Let T_δ={I_1}_i∈  be a regular tessellation of squaresI_iof same length β>0 of^d.  Set𝒢(β)={ i∈:  I_i⊂Ω_0}and note that it is finite due to the compactness ofΩ_0. It is clear that we can writeΩ_0as the union ofΩ^(β)_0=  ⋃_i∈𝒢(β) I_i andΩ^(β,c)_0=Ω∖⋃_i∈𝒢(β) I_i. We define the measureπ_β, =∑_i∈𝒢(β)π_i^χ_(I_i×∇ g^* (I_i)) + γ_β, χ_(Ω_0^(β,c)×Ω_0^(β,c)),where π_i^ ∈Π(χ_I_iρ_0, χ_[∇g^*(I_i)] ρ_1) is the true minimizer of1/2∫-^2dπ(,)+ε/2dπ/  d(ρ_0 ⊗ρ_1) _L^2(χ_I_iρ_0, χ_[∇ g^*(I_i)]ρ_1)^2,among allπ∈Π(χ_I_iρ_0, χ_[∇g^*(I_i)] ρ_1) , and γ_β,  ∈Π(χ_Ω_0^(β,c)ρ_0, χ_∇^2 g^*(Ω_0^(β,c)) ρ_1) to be determined. By construction, π_β, ∈Π(ρ_0, ρ_1) .Moreover, since∇gis the gradient of a convex function and  pushes χ_[∇g^*(I_i)]ρ_1forward toχ_[I_i] ρ_0, it holds that𝒲_2^2(ρ_0,ρ_1)=𝒲_2^2(χ_[Ω_0^(β,c)]ρ_0,  χ_[∇ g^*(Ω_0^(β,c))]ρ_1)+∑_i∈𝒢(β)𝒲_2^2(χ_[I_i]ρ_0, χ_[∇ g^*(I_i)]ρ_1).It also holds that 

    ℍ_ ( π_β, )= ∑_i∈𝒢(β)𝒯_2, , (·)^2(χ_[I_i]ρ_0, χ_[∇ g^*(I_i)]ρ_1)+ 
    ∫-^2 γ_β, (,) + ·γ_β, / (ρ_0⊗ρ_1)_L^2(ρ_0⊗ρ_1)^2.

We define γ_β, following the same steps as π_^(2) in the previous section. Note thatΩ_0^(β,c)satisfies the assumptions of    Lemma <ref>.  Denote 

    W_ = ∫ - ^2  γ_β, (, ) + ·γ_β, / (ρ_0 ⊗ρ_1)_L^2(ρ_0 ⊗ρ_1)^2,

and repeat the calculations yielding (<ref>) and (<ref>). As a consequence,

    lim sup_→ 0W_ - 𝒲_2^2(χ_[Ω_0^(β,c)]ρ_0, χ_[∇ g^* (Ω_0^(β,c))]ρ_1)/^2/d+2≤ C ℓ_d (Ω_0^(β,c)).

Applying Lemma <ref> for each i ∈𝒢(β) yields

    lim sup_→ 0^+ℍ_ (π_β, ) - 𝒲_2^2(ρ_0, ρ_1)/^2/d+2
           ≤d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∑_i ∈𝒢(β)∫_I_i(ρ_0() ρ_1[∇ g^*()]))^-1/d+2ρ_0()+ C ℓ_d (Ω_0^(β,c)) 
           ≤d^d+4/d+2(d+2)^2/d+2/(ℋ^d-1(𝒮^d-1) )^2/d+2∫_Ω∖Ω_0^(β,c)(ρ_0() ρ_1[∇ g^*()]))^-1/d+2ρ_0() + C ℓ_d (Ω_0^(β,c)).
 Theorem <ref> follows by letting β→ 0. 

§ CONCLUSION AND FURTHER PERSPECTIVES

The entropy-regularized transport, without logarithmic terms, has recently been applied to problems such as manifold learning, economics, and biology. In these problems, the goal is generally to find a non-exact but sparse solution to the optimal transport problem (<ref>). In this work, we take a first step towards understanding this problem for quadratic entropy. Additionally, the methodology presented (the stained glass method) can be generalized to more general entropies, unlike the Gaussian approximation method given in <cit.>, which is only valid for logarithmic penalization. We have also observed that, similarly to the problem regularized by logarithmic entropy—where there exists a relationship between the fundamental solution of the heat equation and the solutions of the regularized problem—with quadratic penalization, the solutions can be approximated by a fundamental solution of the porous media equation.

To conclude this work, we will highlight a series of open problems related to entropy-regularized transport with quadratic entropy:


  *  It is known that the solution of (<ref>) has support that contracts as ε decreases, however, the convergence rates of the support are unknown. It is also not known whether the support decreases monotonically.
    
  *  It is known that the optimal potentials of the problem regularized by logarithmic entropy are C^∞ irrespective of ρ_i. For the quadratic case, we conjecture that this may not be the case, and at most, the potentials (solutions of (<ref>)) will have two degrees more of regularity than ρ_i.
    
  *  In the discrete case, it is known that the effect of quadratic norm penalization in a linear program is null for a sufficiently small regularization parameter. However, this value is not specifically characterized in the literature. Indeed, from a mathematical perspective, it would be interesting to study this point regarding the shape of a given polytope.







Set∈∂Ωand the family of balls{𝔹(, 1/n)}_n∈with centerand radium1/n. Set∈Ωand recall that ψ_tends uniformly to0inΩ. Set_0>0such thatψ__𝒞(Ω)≤δ_0. 


Ifdist(, ∂Ω)≥1/n_0, there exists_0=(n_0)such that for≤_0small enough,𝒮_ε,  ⊂Ωand(1-q()) ≥λ∫(ψ_()- C -'^2)_+ℓ_d(')= 1/C  (ψ_())^d+2/2.If dist(𝐱, ∂Ω) < 1/n_0, we denote 𝐱^* ∈min_𝐳∈∂Ω𝐱 - 𝐳. The segment [𝐱, 𝐱^*] is contained in 𝔹(𝐱^*, 1/n_0) ∩Ω; otherwise, 𝐱^* would not be the closest point on ∂Ω to 𝐱. Change coordinates in order to have^*=0and=λe_1withe_1being the first element of the canonical basis.

     Let Ω be an open bounded and set with 𝒞^0,1 boundary with Lipschitz constant L.  There exists n_0∈ and c=c(L)>0 such that for every n≥ n_0 and ∈Ω, 
     ℓ_d( 𝔹(𝐱, 1/n) ∩Ω)≥c/n^d.
It is enough to prove it for ∈∂Ω. 
Assume that 

Let {V_i}_i ∈ I be the finite covering of ∂Ω and φ_i:U_i→ V_i the charts.  For i≠ j, V_i∩ V_j is either empty or there exists an open ball 𝔹(_0, δ_i,j) of radius δ_i,j>0. Let δ^*>0 be the smallest of all of them. Take n_0≥ 1/δ^* and  such that -^*≤ 1/n_0. By construction 𝔹(^*,1/n_0)∩∂Ω⊂ V_i  for some i∈ I.
Set  ^*∈min_𝐳∈∂Ω𝐱 - 𝐳.  Note that  the  segment [𝐱, 𝐱^*] is contained in Ω̅; otherwise, 𝐱^* would not be among the closest points on ∂Ω to 𝐱.  Change coordinates in order to have ^*= 0 and =λ e_1 with e_1 being the first element of the canonical basis. For  t∈ (0, 1/n_0] we define the hyperplane 
    H_t={∈^d: ⟨,  e_1⟩ =t}.
Set _̆t∈∂Ω∩ H_t. Pythagoras theorem applied to the right-angled triangle with vertexes (_̆1,t  e_1, λ e_1) and hypotenuse the segment [_̆1, λ e_1] implies
_̆1 -λ e_1^2=(λ-t)^2+_̆1 -t  e_1^2.  
Since  ^*∈min_𝐳∈∂Ω𝐱 - 𝐳, we have 
2 tλ≤_̆1 -t  e_1^2,
so that 
 dist(t  e, ∂Ω∩ H_t  )≥( 2 tλ)^1/2
The cone 
𝒞_={: arccos(⟨,  e_1⟩/) ≤θ}
We claim that the set
    A={ t∈ (0, λ]: ^*∈min_𝐳∈∂Ω t  e_1 - 𝐳}
   is open and closed in (0, λ) and non-empty. To prove this claim we take a sequence { t_n}_n∈⊂ (0, λ) such that t_n → t and ^*∈min_𝐳∈∂Ω t_n   e_1 - 𝐳.  Then 
    t_n   e_1 - ^*≤ t_n   e_1 - 𝐳  for all ∈∂Ω.
   Taking limits, as n→∞, yields the closeness. 
   
   
   Therefore (t e_1)^*=^* for all t∈ (0,
   Set ∈̆∂Ω∩ H_t and notice that -̆t e_1≥^*-t e_1
    
Take n_0≥ 1/δ^* and  such that -^*≤ 1/n_0. By construction 𝔹(^*,1/n_0)∩∂Ω⊂ V_i  for some i∈ I. 
   Note that  the  segment [𝐱, 𝐱^*] is contained in 𝔹(𝐱^*, 1/n_0) ∩Ω; otherwise, 𝐱^* would not be the closest point on ∂Ω to 𝐱. 
   
   
   
  
   
   
   Set ,̆'̆∈ U_iφ_i()̆
   
   
   Since ∂Ω is Lipschitz and ψ_   tends uniformly to 0 in Ω, we can find _0>0 and a cone 
𝒞_={: arccos(⟨,  e_1⟩/) ≤θ} containing the segment [ 0, λ e_1],  whose angle θ depends only on L and such that
𝒞_∩𝒮_, ⊂𝒮_, ∩Ω, 
for all ≤_0. 


   We know thatV_ε=ε^-d+4/d+2 (f_ε()+g()-⟨, ⟩)_-^2 ρ_1() ρ_0()∫ (f_ε()-g^*()+ ⟨-1/2∇ g^*(),    [D^2g^* ()]^-1 (-∇ g^*())⟩)_-^2 ρ_0() ρ_1() .∫ (f_ε()-g^*()+1/2⟨-∇ g^*(),    [D^2g^* ()]^-1 (-∇ g^*()))_-^2 ℓ_d() ρ_1[∇ g^*()]) ρ_0()Change=̆F()=[D^2g^* ()]^-1/2 (-∇g^*())∫([D^2g^* ()])^1/2∫(f_ε()-g^*()+ ^2/2)_-^2 ℓ_d()̆ρ_1[∇ g^*()]) ρ_0()  .C_d^(2)∫([D^2g^* ()])^1/2| f_ε()-g^*()|^d+4/2ρ_1[∇ g^*()]) ρ_0()  .ε^d+4/d+2 C_d^(2) C_d^-d+4/d+2∫([D^2g^* ()])^1/2 ((ρ_0()ρ_1[∇ g^*()]))^-d+4/2(d+2))ρ_1[∇ g^*()]) ρ_0()  .Since(D^2g^* ())=ρ_0()/ρ_1(∇g()), 



Let Assumption <ref> hold. There exists  δ_0>0 such that for every δ∈ (0, δ_0)  there exist _δ
such that for ≤_δ the following holds: 

  *  There exists a function 
c_, δ such that
q_δ()=1/∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ρ_0(') ,   for all ∈Ω^δ,c_0.   
  *  There exist positive constants C=C(δ_0) and  c=c(δ_0) such that for every  ∈Ω^δ,c_0
    ε^2/d+2 c ≤  c_, δ() ≤ε^2/d+2 C,

and 

    ε^d/d+2 c≤ℓ_d(𝒮_ε,δ, )≤ Cε^d/d+2,

where 𝒮_ε,δ, = {'∈Ω_0:  c_,δ ()-D(, ∇ g^*(')  ≥ 0}.
  *  There exists a positive constant  c=c(δ_0) such that for every  '∈Ω^δ,c_01/∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ρ_0() ≥ c.  We prove each point separately. 

 Proof of <ref>.
Set ∈Ω_0. Since the function 
Θ(a)=1/2 ∫_Ω^δ,c_0 (a-D(, ∇ g^*(') )_+^2ρ_0(')-a q_δ()  
is differentiable,  convex and lim_a→±∞Θ(a)=+∞, there exists a minimizer a() which satisfies 
1/ ∫_Ω^δ,c_0 (a()-D(, ∇ g^*(') )_+ρ_0(')= q_δ() . 
This proves the existence. 

 Proof of <ref>. First we notice that c_, δ() must be positive  and 
c_, δ  tends uniformly to 0 in Ω_0^δ,c. 
Lemma <ref> implies that

    (1-q_δ())
           ≤∫_Ω_0^δ,c(c_, δ()- λ/2-'^2)_+ ρ_0(')
       ≤ C(Λ, c, d) (c_, δ())^d+2/d

and 

    (1-q_δ())   ≥∫_Ω_0^δ,c (c_, δ()- C-'^2)_- ρ_0(')
       ≥λ∫_Ω_0^δ,c(c_, δ()- C -'^2)_+ℓ_d(').

Denote for ∈Ω_0^δ,c the ball 𝔹(,c_, δ()) of center  and radius  c_, δ(). Note that since  ∂Ω_0^δ,c is 𝒞^0,1 with uniformly bounded (for δ≤δ_0 with δ_0>0 small enught) Lipschitz constant, there exist a constant C(δ)=C(δ_0) and _δ such  that 
 C(δ )ℓ_d(𝔹(,c_, δ())∩Ω_0^δ,c)≥  (c_, δ())^d/2. 
As a consequence, using Lemma <ref>, 
   C(λ, δ, g, g^*) ≥    (c_, δ())^d+2/2
and (<ref>) holds. (<ref>) holds due to the nesting 
𝔹(,c  c_, δ())⊂𝒮_ε,δ, ⊂𝔹(,C  c_, δ()),  
which are consequence of Lemma <ref>.

 Proof of <ref>. 
By the previous points, There exists  δ_0>0 such that for every δ∈ (0, δ_0)  there exist _δ and c=c(δ_0)> 0
such that for ≤_δ
    ∫_Ω^δ,c_0 (c_,δ ()-D(, ∇ g^*(') )_+ dρ_0()   ≥ c∫_Ω^δ,c_0 (c_,δ(')- D(', ∇ g()))_+ dρ_0() 
       =c  inf_q_δ()

By Lemma <ref> and letting _δ smaller if necessary,  we can ensure that inf_q_δ()≥ 1/2 for all ≤_δ

 Paper Title: Fisher-Rao distance between truncated distributions and robustness analysis in uncertainty quantification.


Article title]Fisher-Rao distance between truncated distributions and robustness analysis in uncertainty quantification

[1,2]Baalu Belay Ketemabaalu-belay.ketema@edf.fr
2,3,4]Nicolas Bousquet 
2]Francesco Costantino
2]Fabrice Gamboa
1,2,4]Bertrand Iooss
1]Roman Sueur

[1]EDF R&D, 6 quai Watier, 78401, Chatou, France

[2]IMT, Université Paul Sabatier, 118 Route de Narbonne, Toulouse, 31062, France

[3]LPSM, Sorbonne Université, 4 place Jussieu, Paris, 75005, France

[4]SINCLAIR AI Laboratory, EDF R&D, Sorbonne Université, Palaiseau, 91120, France

Input variables in numerical models are often subject to several levels of uncertainty, usually modeled by probability distributions. In the context of uncertainty quantification applied to these models, studying the robustness of output quantities with respect to the input distributions requires: (a) defining variational classes for these distributions; (b) calculating boundary values for the output quantities of interest with respect to these variational classes. The latter should be defined in a way that is consistent with the information structure defined by the “baseline” choice of input distributions. Considering parametric families, the variational classes are defined using the geodesic distance in their Riemannian manifold, a generic approach to such problems. Theoretical results and application tools are provided to justify and facilitate the implementation of such robustness studies, in concrete situations where these distributions are truncated – a setting frequently encountered in applications. The feasibility of our approach is illustrated in a simplified industrial case study.



[
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    August 5, 2024
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§ INTRODUCTION


In applied mathematics, the importance of uncertainty quantification (UQ) stems from the considerable development of engineering and decision support methods. Generally, these methods are based on numerical models or models learned from data (not to mention hybrid ones) and provide predictive diagnoses. Robustness analyses (RA) form an important part of such studies <cit.>.
Indeed, the goal of RA is to quantify the impact of the modeling assumptions in the model input variables.
This step is crucial in many applications, for example when a certification from authorities is required <cit.>.

Very generically, RA  can be  formalized as follows. Denoting G(X)=Y a deterministic model of interest with inputs X:=(X_1,...,X_d) and output Y, X is modeled as a random variable with distribution P_X on ℝ^d (either theoretical or empirical).
Let us define  some quantity of interest (QoI) on Y. For instance, many authors in structural reliability consider a unidimensional output Y and define QoI(Y) as a function of a probability, quantile or superquantile of Y induced by P_X given G <cit.>.  Often P_X suffers from  misspecification.  
Indeed, the assessment of a statistical distribution for X from a mixture of observations and external knowledge, a current approach in many engineering fields <cit.>, is subject to 
modeling errors. 
The stacking of these errors is  generally interpreted as  so-called epistemic uncertainties <cit.>. 
An empirical vision of this problem, in a machine learning framework, is known as the domain shift problem <cit.>. 
The aim of RA  studies is therefore to check, through some robustness indices, whether QoI(Y) remains weakly dependent on this misspecification of P_X, by making P_X vary in a given variational class C, a subspace of the space P of feasible distributions on X.
Therefore, following the semantics proposed by <cit.>, RA studies are based on  distributional perturbation methods.

Defining C was the subject of many works in recent years <cit.>.
A major contribution to the field was made by <cit.>, providing considerable motivation for our work. In this paper, the inputs X_1,...,X_d are assumed independent and P_X varies only through its d marginals P_X_i in a certain family 𝒫_i of distributions on ℝ. The authors then argue that a generic RA with respect to (w.r.t.) input domain misspecification requires to handle the information geometry of each 𝒫_i 
through its Fisher-Rao (FR) distance d_i <cit.>. This requires that each 𝒫_i is 
a parametric family 𝒫_i={P_X_i,θ}_θ∈Θ_i of distributions on ℝ containing P_X_i and with well-defined Fisher information. The authors also explain that a good candidate for the variational class 𝒞 is given by a collection of concentric FR spheres Λ_iδ with radii δ∈(0,δ_max] centered at the marginal distribution P_X_i, for each i. Then, they propose to compute robustness indices based on the maximum and minimum values of a QoI over these FR spheres Λ_iδ.  

This approach promotes a more objective RA framework than the perturbations of probability distributions based on entropy criteria,  that involve inherently subjective form constraints, as generalized moment constraints proposed in <cit.>. Besides, perturbations based on the intrinsic geometry of distributions are by essence multivariate, which allows to go beyond RA based on optimal transport techniques modifying marginal percentiles  <cit.>. 

The proposal made by <cit.> is part of a slow but progressive trend in UQ to manipulate Riemannian and information geometry.
Information theory was used in <cit.> as early as 2008 
to define sensitivity indices based on input-output mutual information. One of the first use of information geometry in UQ may be found in <cit.> for building reduced order models from snapshots. Other works in this field followed since, for instance in Bayesian UQ <cit.>. Very recently, a symplectic decomposition of the Fisher information matrix is used for parameter sensitivity analysis in <cit.> while information geometric techniques are put in action for studying sloppy models in <cit.>. But up to our knowledge, the only work proposing an information geometry-based method for RA applied to numerical models, handling two-parameter distributions (but suffering from poor accuracy performance linked to Monte Carlo approximations of Fisher information) remains <cit.>. 

This relatively slow adoption by the UQ community is probably due to: (a) a lack of knowledge about the required geometrical tools; and (b) a lack of known results (except those produced in <cit.>) concerning the theoretical features and practical calculations, for concrete problems, of parametric distributions defining C.


This twofold need is the main motivation of our work. Let us discuss the results below. First, we provide an illustration of the  method and highlight the explicit correspondence between the FR spheres Λ_iδ  and distributional perturbations on selected families of 
probability distributions (normal, log-normal, Gumbel, Gamma, triangular, exponential, etc.).  Second, we obtain new explicit computations on Fisher information and Christoffel symbols for a truncated version of the normal distributions. This is of particular interest as they are often used in applications <cit.>. This allows us to derive the corresponding FR distances. Lastly, we  present more general results for handling two common classes of families of distributions, namely location-scale and pushforward families. 



More precisely, we obtain these results by taking the following path. We first propose in Section <ref> an initiation or reminder on information geometry, in a classical theoretical setting. 
This description is adapted to the special case of truncated distributions in Section <ref>, where we also detail the RA framework, explain the distributional perturbation method and illustrate the latter on the normal family in usual and truncated cases. Section <ref> gives the main algorithm for computing the FR spheres Λ_δ and illustrates the perturbation method on the families mentioned above. A numerical experiment is proposed in Section <ref>, describing the 
optimization procedure and illustrating the feasibility of robustness analyses, through the study of a simplified flood model. Besides, a discussion section closes our paper, opening towards several research perspectives aimed at extending the use of this type of robustness analysis in UQ problems.

We postpone to Appendix <ref> some technical details on the statistical procedure for estimating (with bootstrap-based confidence intervals) the robustness indices proposed in <cit.>. Proofs and additional geometric explanations are given in Appendix <ref>. Some tedious technical computations and explanations on the Fisher information and Christoffel symbols are provided in a Supplementary Material. 
Finally, Table <ref> provides the main acronyms used in this paper.





§ RIEMANNIAN GEOMETRY OF PROBABILITY DISTRIBUTIONS


Since the pioneering works by Hotelling <cit.>, Rao <cit.> and Jeffreys <cit.>, exploring the geometry of parametric families of distributions has become a very active reseach area, 
which has notably produced new interpretations of the mechanisms of statistical inference <cit.>. However, this framework often remains unfamiliar to UQ practitioners. 
Therefore basic concepts and tools of Riemannian and information geometry – which can be skipped on first reading – are briefly presented in the following paragraphs. Readers interested in more details are referred to  <cit.>, <cit.>, <cit.> and <cit.>.



 §.§ Basic notions


Riemannian manifolds are smooth manifolds M equipped with a metric tensor usually denoted g (page 38 in <cit.>, Definition 2.1). At a given base point p∈ M, this metric tensor denoted g_p is a scalar product on each tangent space T_p M of M and varies smoothly w.r.t. the base point p. This metric allows to generalize on manifolds many  geometric notions like angles, norms and distances already known on the 2D plane or 3D space. For instance, the distance between two points  and  on M is defined as the length of the shortest paths connecting  and 
    d(,):= inf_α(0) = ,  α(1) = ℓ(α)

where ℓ(α) is the length of the curve α:[0,1] → M defined as

    ℓ(α) = ∫_0^1 |α̇(τ)|_α(τ)  dτ

and |v|_ is the norm of v∈ T_ M defined as |v|_= √(g_(v,v)). This distance is called the Riemannian geodesic distance and it is intrinsic to the metric g. 

Affine connections, an important class of objects in geometry, are differential operators that allow to compute directional derivatives of vector fields. On Riemannian manifolds there exists a unique one, called the Levi-Civita connection and denoted ∇̅, that is compatible to the metric (and torsion-free). Roughly speaking, the connection ∇̅ provides a natural extension of the classical formula for differentiating a scalar product (see Definition 3.1 page 53 in <cit.>). Denoting m the dimension of M, at each point p∈ M this connection is encoded by m^2(m+1)/2 coefficients Γ̅_ij^k(p) for 1≤ i,j,k≤ m called Christoffel symbols, which depend on a local chart around p, say x^1,...,x^m. In this chart, these symbols are given by

    Γ̅_ij^k  = ∑_l=1^mg^kl/2( ∂ g_il/∂ x^j + ∂ g_jl/∂ x^i - ∂ g_ij/∂ x^l),
  
where g_ij and g^ij correspond resp. to the ij coefficient of the matrix of g and its inverse and ∂ g_ij/∂ x^l are the partial derivatives of g_ij  in the chart x^1,...,x^m. All these quantities are evaluated at point p∈ M.

Affine connections allow us to define the second derivative, or “acceleration", of a curve γ: [0,1] → M 
by seeing t ↦γ̇_t as a vector field along γ and computing its directional derivative at point γ_t and in the direction γ̇_t as

    ∇̅_γ̇_tγ̇_t ∈ T_γ_t M,

which can be thought as the derivative of γ̇ i.e. the second derivative of γ. Curves that have no “acceleration" are called geodesics and they solve a second order differential equation given by ∇̅_γ̇γ̇= 0, or in coordinates (with Einstein summation convention)

    γ̈^k + γ̇^i Γ̅_ij^k γ̇^j = 0,   for all  k=1,...,m.
  

Geodesics γ locally minimize the Riemannian distance (<ref>) since for small t>0 

    d(γ_0,γ_t) = ∫_0^t |γ̇_τ|_γ_τ  dτ = t|γ̇_0|_γ_0.

This shows that  
geodesics γ are constant speed curves,  
since the distance between γ_0 and γ_t is given by the initial speed |γ̇_0|_γ_0 multiplied by the travel time t.



 §.§ Geometry on a family of parametric probability distributions

Given a parametric family 𝒫 = {P_θ}_θ∈Θ 
of probability measures, with Θ an open set in ℝ^d, the seminal idea proposed by Hotelling and Rao <cit.> is to describe 𝒫 as a Riemannian manifold endowed with the Riemannian metric given by the Fisher Information matrix (FIM) I_θ, whose components are 

    (I_θ)_ij = 𝔼_X∼ P_θ[∂_i log p_θ(X)∂_j log p_θ(X)]

where p_θ = dP_θ/dμ is a probability density function (pdf) of P_θ w.r.t. a reference measure ν and ∂_i log p_θ(x) = ∂/∂θ^ilog p_θ(x).

A few regularity conditions are usually assumed to ensure the FIM exists and is positive semi-definite. For instance (P_θ) should be independent from θ∈Θ;  
the map θ↦ p_θ(x) should be smooth;  and 
the integration 𝔼_X∼ P_θ(·) and differentiation ∂/∂θ^i operations should be interchangeable. 
In addition, linear independence of the functions {x ↦∂_i p_θ(x)}_1≤ i≤ d for all θ guarantees the positive-definiteness of I_θ. 
More details on the required regularity assumptions are provided in <cit.>. 

Under these assumptions, the FIM I_θ is the matrix representation of a scalar product g_θ on the tangent space of 𝒫 at P_θ, usually called Fisher metric. 
Assuming the smoothness of (<ref>) w.r.t. θ, the couple (𝒫,g) becomes a Riemannian manifold. This allows to conduct geometrical analyses on 𝒫 using the intrinsic Riemannian distance (<ref>), geodesic curves (<ref>) and many more tools.
With our previous notations, the manifold is M:=𝒫 and the metric tensor is g_θ.

For a parametric family of distributions 𝒫, the Riemannian geodesic distance d given by (<ref>) is called the FR distance. This distance is intrinsic to the family 𝒫 and is invariant under reparameterization 
<cit.>. In addition, for P_θ in the vicinity of P_θ_0, the FR distance behaves like the Kullback-Leibler divergence 𝒟: 

    𝒟(P_θ | P_θ_0) = 1/2 d(P_θ,P_θ_0)^2 + o(d(P_θ,P_θ_0)^2),

where 𝒟(P_θ| P_θ_0):= 𝔼_X∼ P_θ[log(P_θ/P_θ_0(X))]. However, a global statistical interpretation for d is less obvious. 
Computing an approximation of d is still possible if the Christoffel symbols (<ref>) are available. This will be further discussed in Section <ref>.





§ ROBUSTNESS ANALYSIS AND TRUNCATED PROBABILITY DISTRIBUTIONS




 §.§ Robustness analysis framework 

Echoing the concepts and notations introduced in the first section, let us consider the RA framework <cit.> defined by:

    
  * A QoI on the stochastic variable Y:=G() where G is a deterministic forward model and
    = (X_1,…,X_d) is a random vector with distribution P_;
    
  * A family 𝒫_i, for each input X_i, of probability distributions each containing the baseline marginal P_X_i of P_, with pdf f_i;
    
  * A distance d_i on each family 𝒫_i, that will be precised in the sequel. This allows to define the δ-perturbations of f_i as any other element  f_iδ∈𝒫_i at d_i-distance δ from f_i. The collection of such elements is denoted by Λ_iδ and forms a metric sphere in 𝒫_i; 
    
    
  * A set {S_i}_1≤ i ≤ d  of robustness indices, as for example the perturbed law-based indices (PLI ; <cit.>) defined for nonzero QoI:
    
    S_i(f_iδ)  =  QoI(Y^iδ)/QoI(Y) - 1,

    where Y^iδ:=G(X^iδ) for X^iδ∼ P_X^iδ and P_X^iδ is the joint distribution of  for which only the i-th marginal pdf f_i has been replaced by some f_iδ∈Λ_iδ. In other words, P_X^iδ has density f_1⊗…⊗ f_iδ⊗…⊗ f_d.


Having set the RA framework, we now explain how the RA study of G is converted into a mathematical problem. We then precise the distance d_i used on 𝒫_i before defining truncated distributions and explaining our interest in such families for RA. The δ-perturbations for distance d_i are illustrated in the normal family for both the truncated and non-truncated case. 

The indices S_i can be seen as functions f_iδ∈𝒫_i ↦ S_i(f_iδ) ∈ℝ quantifying the robustness of QoI(Y) w.r.t. perturbations on f_i. A value close to zero for S_i(f_iδ) means that QoI(Y) is not significantly affected when perturbing f_i to f_iδ, reflecting its robustness w.r.t. the distribution of X_i. The following bounds or synthetic indices, introduced in <cit.>, quantify the worst impact of such perturbations given a δ ranging in (0,δ_max]:

    S_iδ^-:=min_f_iδ∈Λ_iδ S_i(f_iδ)     and   S_iδ^+:=max_f_iδ∈Λ_iδ S_i(f_iδ) ⋆.

Ultimately, a maximum level δ_max must be elicited to get a practically usable methodology, which remains an open topic (see Section <ref>).

In practice, the robustness indices {S_i(f_iδ)}_i cannot be computed explicitly
and must be estimated using evaluations from the model G. Details on the estimation procedure used in <cit.> are provided in Appendix <ref>. Similarly, the resolution of (⋆) cannot
be achieved through usual gradient-based optimization techniques and thus requires a dedicated
algorithm.



 §.§ Defining the perturbation sets 

 


Defining how the baseline pdf (or equivalently distribution) f_i should be  perturbed has been a research subject considered by numerous authors <cit.>. 
In this article, for the sake of genericity and following the recommendations given in <cit.>, we assume 𝒫_i is a parametric family 𝒫_i={f_i,θ}_θ∈Θ_i containing f_i.
Each 𝒫_i is endowed with the Fisher metric as explained in Section <ref> and d_i is taken as the corresponding FR distance. Accordingly, the subsets {Λ_iδ}_δ≤δ_max are defined as concentric FR spheres centered at f_i with radii δ∈ (0,δ_max]:

    Λ_iδ:= {f_i,θ∈𝒫_i |  d_i(f_i,f_i,θ)=δ}.


Let us illustrate this distributional perturbation method on 𝒫:={𝒩(μ,σ^2)}_(μ,σ)∈Θ the normal family by computing the spheres Λ_δ, the exact method for such computations is detailed in Section <ref>. Let us first point out that the FIM in the (μ,σ)-parameterization is easily obtained (see <cit.>). Indeed, we have

    I_θ = 1/σ^2[ 1 0; 0 2 ],    θ=(μ,σ) ∈Θ,

which defines the same geometry as the so-called Poincaré metric on the Poincaré half-plane ℍ:={x+iy  |  y>0} (see <cit.>, as well as Section <ref> below on location-scale families). Most geometric quantities can thus be computed explicitly. This is not the case in general. For instance the Christoffel symbols Γ̅_ij^k (<ref>) are given by

    Γ̅_12^1=Γ̅_21^1 =Γ̅_22^2= -1/σ,    Γ̅_11^2=1/2σ,    Γ̅_11^1 = Γ̅_22^1 = Γ̅_21^2 = Γ̅_12^2 = 0.


Figure <ref> gives an illustration of the concentric FR spheres and δ - perturbations of a normal density function.





 §.§ Robustness analysis for truncated distributions


Specifying the RA framework for truncated marginals of P_X appears particularly 
relevant in many engineering problems (e.g. in risk, reliability and safety issues <cit.>). 

In UQ problems, uncertain (stochastic) inputs of computer models G used for such analyses are often considered on a restriction of their support, focusing on the most critical situations (i.e., leading to critical output values Y). For instance, in <cit.> the authors consider two case studies in flood and nuclear risks, each involving computer models where the input distributions (e.g., Gumbel, normal, log-normal) are truncated on sub-domains that are most likely to lead to undesired output situations. Such studies, that typically aim to test designs, are obviously concerned with robustness analyses. Other truncated distributions can originate from the physical limitations of sensors <cit.> or to respect the physical meaning of input variables (e.g., in hydrology <cit.>, wind energy production <cit.> or wind lidar calibration <cit.>).    

Therefore, we focus on a 1D marginal component X of , and consider interval truncation. 
Assuming that the possible distributions of X are modeled by a parametric family 𝒫:={
P_θ}_θ∈Θ with densities f_θ, the truncated distributions of X∼ P_θ on an interval [a,b] are defined as the conditional probability distributions Q_θ of X∼ P_θ over [a,b] with corresponding pdf q_θ resp. given by

    Q_θ(A):=P_θ(A∩[a,b])/P_θ([a,b])    and    q_θ(x) := f_θ(x) /P_θ([a,b])1_x∈ [a,b].

We can therefore define the FIM, the FR distance and other geometric quantities on the truncated family 𝒬:={Q_θ}_θ∈Θ.

Note that beyond the interval truncation considered here, truncated marginal distributions can be defined on any measurable subset B⊂ℝ as well as for densities on ℝ^d.

Before moving on to the next section, let us illustrate this by considering a truncated version of the normal family on [a,b], denoted 𝒩_[a,b], whose pdfs are given by


    q_θ(x) = 1/N_θ1/√(2π)σexp(-(x-μ)^2/2σ^2)1_x∈[a,b],    for θ = (μ,σ),

where N_θ=(√(2π)σ)^-1∫_a^b  exp(-(x-μ)^2/(2σ^2)) dx is the normalizing constant. We proved  the following result. 

 
    The FIM for the 𝒩_[a,b] family is given by

    I_θ^[a,b] = 1/σ^2[                                      ∂_μμ_[a,b]                                      ∂_σμ_[a,b];                                      ∂_σμ_[a,b] σ^-1( ∂_σσ_[a,b]^2 + 2(μ_[a,b] - μ) ∂_σμ_[a,b]) ],

where μ_[a,b] and σ_[a,b]^2 are functions of θ=(μ,σ) given resp. by the expectation and variance of q_θ.

The exact computation of the FIM coefficients and Christoffel symbols 
is detailed in the online supplementary materials. 
FR Spheres Λ_δ and δ-perturbations in the truncated family 𝒩_[-2,2] are illustrated in Figure <ref>.



We observed that the spheres in the truncated family on the left of Figure <ref> are more distorted, compared to the spheres of the non-truncated normal family in Figure <ref>. We also saw that, numerically for the truncated pdf 𝒩_[-2,2](0,1), a radius δ larger than 0.5 seems to give a non-compact FR sphere. This surprising phenomenon is discussed in Section <ref> of the Appendix in a more general context. 

Furthermore, on the right of both figures, we show the δ-perturbations densities from the nominal density, given a value for δ. They represent distributions that are in the vicinity of the latter for the FR distance. For the truncated pdfs, the δ - perturbations of 𝒩_[-2,2](0,1) are quite distinct: some have been affected by the truncation procedure, while not much for others. In addition, some have flattened and are close to the uniform distribution on [-2,2].





§ ILLUSTRATION OF THE DENSITY PERTURBATION METHOD AND FISHER-RAO SPHERES
 

For the sake of clarity, we illustrate here the nature of δ-perturbations for  some well known parametric families, in usual and truncated situations. These δ-perturbations are obtained by computing the FR spheres Λ_δ in 𝒫_i={f_i,θ}_θ∈Θ, that constraints the optimization problem (⋆). Let us first state the following classical result from Riemannian geometry.
 
Let (M,g) be a Riemannian manifold. The sphere 𝒮(p,δ) for the geodesic distance centered at p ∈ M with small enough radius δ>0 can be identified as
    
    𝒮(p,δ)={γ(1)  |  γ geodesic s.t. γ(0)=p, γ̇(0)=v  and  |v|_p=δ}.

     

This result is not always true for large radii, see Appendix <ref> for more details and Figure <ref> for an illustration of a possible outcome. However, Proposition <ref> allows us to discretize small geodesic spheres through the following algorithm, valid in any Riemannian manifold, where K is the number of discretization points 

Hence for a parametric family 𝒫_i={f_i,θ}_θ∈Θ, we can apply this algorithm in the coordinate chart Θ to obtain K points θ_ℓ∈Θ such that {f_i,θ_ℓ}_ℓ=1,...,K discretizes the sphere Λ_δ in 𝒫_i.
In practice, since we mainly consider two-parameter families, the K tangent vectors v_ℓ with norm δ are taken at uniform angle i.e. proportional to the vectors (cos(2πℓ/K),sin(2πℓ/K))^⊺, ℓ=1,…,K. 

For each two-parameter family considered below, 
we computed the FIM and the Christoffel symbols Γ̅_ij^k (<ref>), and applied Euler method to solve the geodesic equation (<ref>). This  slightly differs from the Hamiltonian method used in <cit.>. When the Christoffel symbols are not known explicitly (Gumbel and Gamma), we approximated them using (basic) numerical integration and differentiation methods.



 §.§ Triangular distributions 

The family 𝒯_[a,b] of triangular distributions supported on [a,b] contains the densities q_m given by 

    q_m(x)= 2(b-a)^-1[(x-a)(m-a)^-11_x∈ [a,m] + (b-x)(b-m)^-11_x∈ (m,b]],   m∈ [a,b].

The Fisher information is given by
i_m = [(b-m)(m-a)]^-1  for m ∈ (a,b).
Since 𝒯_[a,b] is a one parameter family, the FR distance d can be easily computed:

    d(q_m_0,q_m_1) = |∫_m_0^m_1√(i_m) dm| = | arcsin(m_1-α/√(β)) - arcsin(m_0-α/√(β))|,

where α:=a+b/2 and β:=(a-b/2)^2. This allows to explicitly compute the FR spheres (see Appendix <ref>).
Note that the radius cannot be too large as 𝒯_[a,b] has finite diameter

    diam(𝒯_[a,b]) := sup_m_0,m_1 ∈ [a,b] d(q_m_0,q_m_1) = |arcsin(-1)-arcsin(1)| = π.

Hence, the radius δ satisfies δ≤δ_max=π/2. 
Figure <ref> gives an illustration. 




The family 𝒯_[a,b], being a (connected[A connected manifold is a set that is not formed by two or more disjoint sets i.e. it is in one piece.]) 1-dimensional Riemannian manifold, is isometric[Two spaces are isometric if there exists a smooth invertible map between them that preserves the Riemannian structure.] to an Euclidean (straight) interval, of which there exists three types: a half line [0,∞), a finite interval [0,ℓ] (for some positive ℓ∈ℝ) or the whole line ℝ. The family 𝒯_[a,b] corresponds to [0,π] since it has a diameter of π.




 §.§ Location-scale families

Location-scale families are two parameter families obtained by translating and scaling an initial smooth probability density p: ℝ→ (0,∞). Let m∈ℝ and s>0, then

    p_θ(x) = 1/s p(x-m/s),     θ=(m,s).

Here, m and s are resp. the location and scale parameters. 
Many well known families fall in this model. For instance, this is the case for the normal, Gumbel and Cauchy distributions. The following proposition shows that all location-scale families share a common geometry. 


    The FIM of a location-scale family is given by
    I_θ = [ α/s^2 γ/s^2; γ/s^2 β/s^2 ],
    where α,β,γ do not depend on θ=(m,s) and are given by integral expressions involving p and p'. 
    In addition, there exists a linear reparameterization with Jacobian matrix P^⊺ such that
    
    I_θ = P·(1/s^2[ 1 0; 0 1 ]) · P^⊺.

    Hence, we find that the Fisher metric of location-scale families is nothing more than the Poincaré metric.


The proof of Proposition <ref> is based on the spectral theorem. 
We also refer to <cit.> for a proof dealing with an even density p:ℝ→ (0,∞). Up to our knowledge, concluding the same for truncated location-scale families is an open problem.

Numerical computations of FR spheres for the Gumbel family in the truncated (Gumb_[a,b]) and non-truncated (Gumb) situations, as well as the δ-perturbations for both, are illustrated in Figure <ref>. 







 §.§ Bijective push-forward of a parametric family


Here, we work with a family of probability distributions obtained by pushing forward a reference family 𝒫 :={P_θ}_θ∈Θ on some space 𝒳 to 𝒴 by a smooth invertible map 
    h:𝒳→𝒴.
 
For instance, if we take h(x) = e^x and 𝒫 to be the family of normal distributions, then h_#𝒫 := {h_# P_θ}_θ∈Θ becomes the log-normal family. Here h_# P_θ denotes
the push-forward distribution. Set 

    R_θ := h_# P_θ

and let ℛ:= {R_θ}_θ∈Θ be the push-forward family. Our goal is to determine the FIM of a truncated version of ℛ on some subset of 𝒴. Let us first remind the following theorem
 If 𝒳,𝒴 are open subsets of ℝ^d and h:𝒳→𝒴 is smooth and invertible, then for any regular parametric family 𝒫 on 𝒳 the pushforward family ℛ:=h_#𝒫 has the same FIM as 𝒫.


This result directly implies that geodesics and FR spheres in 𝒫 are the same in ℛ (i.e., they are isometric). As an example, this holds for the normal and the log-normal families.

Now, we would like to compute the FIM of a truncated version of ℛ. Let us first fix some notations. From now on, we denote B⊂𝒴 the truncation domain (a measurable subset) for ℛ and  A:= h^-1(B)⊂𝒳 its preimage. 
We denote the truncated family of 𝒫 on A (resp. of ℛ on B) as 𝒫_A :={P_θ^A}_θ∈Θ
(resp. ℛ_B :={R_θ^B}_θ∈Θ). 
In addition, let I_θ^A (resp. K_θ^B) denotes the FIM of 𝒫_A (resp. of ℛ_B).

Since we want to compute the FIM of ℛ_B={(h_#P_θ)^B}_θ∈Θ, if we apply the previous theorem on 𝒫_A={P_θ^A}_θ∈Θ with the restriction of h to A, we get that 𝒫_A and 


    h_#𝒫_A := {h_#(P_θ^A)}_θ∈Θ

have the same FIM. 

The following result states that h_#𝒫_A and ℛ_B are the same, 
the proof being obvious and intuitive. 

Under the previous notations, it comes  
    (h_#P_θ)^B = h_#(P_θ^A),     for all θ in Θ.

Therefore ℛ_B and h_#𝒫_A are the same family.
Moreover, ℛ_B and 𝒫_A have the same Riemannian structure since their FIM coincide:

    I_θ^A = K_θ^B,     for all θ in Θ.



This result allows us to deduce K_θ^B, the FIM of the truncated push-forward family ℛ_B on 𝒴, if we know I_θ^A, the FIM of the initial truncated family 𝒫_A on 𝒳 where A =h^-1(B), see Example <ref>. The following diagram summarizes our considerations:



(𝒳, P_θ) [r, "tr. on A"] [d,  " h_#"]        (𝒳, P_θ^A) [d, " h_#"] 

(𝒴, R_θ) [r, "tr. on B" ]         (𝒴,R_θ^B)



The previous lemma tells that the push-forward operation and the truncation operation commute. In addition, the standard push-forward theorem shows that the left-down arrow preserves the Fisher information, and the lemma guarantees that the Fisher information is preserved by the right-down arrow.

 
    For the family log𝒩 of log-normal distributions
    
    r_θ(x) =1/xσ√(2π)exp(-(log x -μ)^2/2σ^2)1_x>0,     for θ = (μ,σ),

    Lemma <ref> implies that the FIM of log𝒩_[a,b], the truncated version of this family on [a,b], is given by
    I_θ ^[log a, log b],
    where I^[α,β] is the FIM of 𝒩_[α,β] provided in Proposition <ref>. In particular, the Levi-Civita Christoffel symbols of 𝒩_[log a,log b] and log𝒩_[a,b] are identical 
    as well as their geodesics and FR spheres on the upper halfplane. Figure <ref> gives an illustration of this result.




A natural question would be to know whether the previous lemma still applies when h:𝒳→𝒴 is a sufficient statistic for the family 𝒫. Indeed, in this case, the Fisher metric of 𝒫_A and ℛ_B are the same. The proof, which is obvious, is provided in Appendix <ref>.



 §.§ Exponential families


Exponential families are wide families of probability distributions, with normal, log-normal, gamma, binomial, Poisson and geometric distributions amongst their most notable members. The densities f_θ of an exponential family with reference distribution ν on ℝ^m are given, for θ∈Θ, by

    f_θ(x)=e^x·θ - ψ(θ),    x∈ℝ^m, θ∈Θ.

Here, we denote x·θ the Euclidean scalar product on ℝ^m, the function ψ is the log-Laplace transform of ν and Θ is the domain of ψ. The FIM for exponential families is given by the Hessian of ψ <cit.>, hence exponential families can be seen as Hessian manifolds <cit.>.

A truncated exponential family is built from an initial exponential family truncated on an interval [a,b], here we take m=1. These truncation bounds are usually considered as parameters <cit.>, and therefore these families are not regular and the FIM is not necessarily well defined. In this subsection, we consider truncated exponential families on an interval [a,b] where a,b are fixed  – a special case of truncated distributions studied by <cit.>. 

The truncated densities with normalizing constant N_θ also form an exponential family since

    N_θ^-1f_θ(x)1_x∈ [a,b]dν(x) = e^x·θ - ψ(θ) - log N_θdν_[a,b](x),

where ν_[a,b](·):=∫_·1_[a,b] dν is the reference distribution and ψ̃(θ):= ψ(θ) + log N_θ is the log-Laplace transform of ν_[a,b]. Hence the FIM of this family is given by the Hessian of ψ̃. 
See Figure <ref> for illustrations in the non-truncated case, and refer to Appendix <ref> and the Supplementary Material for details on computations.






§ NUMERICAL EXPERIMENTS: ROBUSTNESS STUDY OF A FLOOD MODEL



In this section, the RA methodology discussed in Section <ref> to is applied for studying the robustness of an analytic flood model. This model has been studied in previous articles such as <cit.>. The code we developed is available at https://github.com/baalub/Truncated-distributions.githttps://github.com/baalub/Truncated-distributions.git





 §.§ Context

The flood model computes a river's maximal annual water level H assuming constant and uniform flow rate. This is useful information for predicting if flooding will occur. The model is defined as 

    H := G(K,Q,Z_m,Z_v) = Q^0.6(300K √(Z_m - Z_v/5000))^-0.6,

where Q∈ [0,3000] is the maximal annual flow rate, K ∈ [15,90] is a roughness (Strickler-Manning) coefficient and Z_m ∈ [54,56] and Z_v ∈ [49,51] are resp. the upstream and downstream heights of the river. Figure <ref> gives an illustration of the river and Table <ref> summarizes the baseline distributions, selected from legacy data and  expert knowledge.






The output H corresponds to the river depth and flooding will occur if H + Z_v≥ Z_d, where Z_d is the height of the dyke. The QoI we look at is the 90% quantile of the output H.
We perform the RA method presented above to determine which inputs are robust w.r.t. input density perturbations. As indicated in Section <ref>, we compute 

    min_f_iδ∈Λ_iδS_i(f_iδ)     and   max_f_iδ∈Λ_iδS_i(f_iδ) ⋆⋆

where S_i is an estimator of S_i (see Appendix <ref> for details), i∈{Q,K,Z_m,Z_v} and δ∈{0.1,0.2,...,1}. Here, we picked δ_max=1 for practical reasons. Three-dimensional plots of the quantile estimator on the concentric FR spheres are provided on Figure <ref>. Besides, Figure <ref> highlights the maximum and minimum values of S_i on concentric FR spheres, as a function of δ. The confidence intervals accompanying this result are built using bootstrap techniques, detailed in  Appendix <ref>. 







 §.§ Optimization procedure

Computing the bounds in (⋆⋆) for the triangular inputs Z_m and Z_v was not challenging since in these cases the FR spheres Λ_iδ were made up of two points. For the truncated normal K and truncated Gumbel Q inputs, our approach for computing (⋆⋆) is to discretize each sphere Λ_iδ, for δ∈{0.1,0.2,...,1}, into 100 points. We then optimize the estimator S_i of the robustness index. We use this approach because the estimator S_i is piecewise constant (see Appendix <ref>) and thus a gradient-based method cannot be used. Improving this optimization procedure is a subject of current work, evoked further in the Discussion section.  

Discretizing the concentric FR spheres takes around 8 minutes for each input K and Q. Evaluating S_i on each point of the discretization (1000 points) for a sample of size N=10^4 took around 3 hours in total for all the inputs K,Q,Z_m,Z_v on an 11^th gen Intel(R) Core i5-1135G7 @2.40Hz computer. The time required for the estimation of the robustness index highly depends on the choice of the QoI. Usually, estimating the expectation or a probability threshold of the output takes significantly less time than a quantile or a superquantile.



 §.§ Analysis of the numerical results


From Figure <ref>, we observe that the output of the model G is sensitive to density perturbations on K and Q of level of at most δ_max = 1. For the inputs Z_m and Z_v, it seems that perturbing their initial densities does not affect much the output.

Figure <ref> highlights that as the perturbation level δ increases a minimizer of S_Q has a tendency to give more weight to points in [0,3000] that are close to 0. 
This is not surprising since G is increasing as a function of Q. 







This RA result for the flood model is comparable to previous RA studies in papers such as <cit.>. Indeed, in this paper, the inputs Z_m and Z_v were also the most robust to input density perturbations. However, in contrast to our case, the input Q was the least robust in <cit.>. This may be due to the difference in the baseline distributions for Q and K.



§ DISCUSSION
 

In this paper, we have explained the RA methodology based on the FR distance on families of probability distributions. This distance allowed us to define δ - perturbations of densities in some parametric family. We then illustrated the feasibility of this δ - perturbation method in some classical families of distributions both in the truncated and non-truncated case. This required either an explicit formula for the FR distance or an approximation of the Christoffel symbols of the corresponding Fisher metric. This showed that the FR distance, which might seem hard to compute at first glance, can be handled rather easily and that the FR spheres can be approximated with good precision (at least for small radii). We also proved a result that allows us to deduce the Fisher metric for some truncated families obtained by a bijective push-forward operation of another truncated family (Lemma <ref>). Finally, we applied the RA method to an analytic flood model to illustrate its applicability.

Compared to the previous RA methods, the advantage offered by this approach is that it gives a mathematically rigorous and objective way for defining density perturbations. Nevertheless, the method leads to obstacles that have to be overcome in practice.

First, the optimization method used for problem (⋆⋆) requires to discretize the FR spheres into finite points and then to optimize S_i on these points. 
The choice for such an optimization procedure is related to the estimation method for S_i, but obviously it does not scale well to higher dimensions. One solution would be to regularize S_i and perform Riemannian gradient descent on the FR spheres. Moreover, one can also consider the same optimization problem constrained to the FR ball with radius δ_max instead of on each concentric spheres, which reduces to solving a single optimization problem and allows to obtain the maximum and minimum value of S_i for all δ≤δ_max.

Second, we only considered the case where the QoI on the output Y is a quantile for the flood model. However, QoIs for which (⋆) becomes an expectation optimization problem can be directly tackled with stochastic optimization algorithms, which allows to avoid optimizing an estimator of the objective function. This will be the topic of a future work.

Another central question is obviously to set a maximum value δ_max in practice. To do so, let us first note that this maximum value can be imposed by the intrinsic geometry of the family of distributions (see the example of the family of triangular distributions). In contrast, other families with infinite diameter (location-scale families for instance) do not impose such constraints and spheres of arbitrarily large radii are well defined. Furthermore, some families with infinite diameter can contain spheres that are not compact. This may indicate the existence of distributions outside the family that are at finite FR distance. Thus, this can be useful for defining a possible δ_max at a density f_θ_0

    δ_max:= sup{δ>0  |  𝒮(f_θ_0,δ)  is compact},

where 𝒮(f_θ_0,δ) is the FR sphere at f_θ_0 with radius δ. 

This non-compactness property of spheres is linked to geodesic completeness of the family, through the Hopf-Rinow theorem <cit.>. 
Up to our knowledge, completeness of families of distributions is not well studied in the literature (even less for truncated ones). Hence determining an intrinsic δ_max in this way for a specific family is a challenging open problem. Actually δ_max is only defined for a point in a specific family. Note further that compactness of FR spheres Λ, coupled with a continuity assumption on f_iδ∈Λ↦ S_i(f_iδ), is useful for guarantying existence of solutions to (⋆). 

Moreover, choosing a δ_max should also ensure that distributions within the FR sphere defined in this way have “practical" plausibility with regard to the UQ problem. For example, we could try to impose that f_iδ_max should correspond to the least informative plausible distribution (e.g., a uniform distribution) or the most penalizing possible distribution (in the sense of least favorable prior distributions in the Bayesian framework <cit.>).

Another approach, extrinsic to the geometric problem and which would enable us to compare different perturbations approaches for RA, could be to interpret δ_max as the result of an inverse problem. With a view to testing robustness against a limit risk defined, for example, as a limit output α-quantile q_lim(α) for Y, with 0<α≪ 1, as in <cit.>, consider

    δ_i,max    =    min_δ_i|q_lim(α) - q(α ; δ_i) |

where q(α ; δ_i) is the α-quantile of Y_i=G(X_1,…,X_i,δ_i,…,X_d) where the X_j∼ f_j for j≠ i and X_i,δ_i∼ f_iδ_i. Such a definition  would give a practical, model-agnostic and more easily interpretable meaning to the different values of δ_max. It would, however, require the resolution of d inverse problems. This could be computationally time-consuming. 

While the methodology has been illustrated in situations where the components of  are independent, it is easily adaptable to cases of stochastic dependence modeled by copulas. These are amongst usual implementation choices in UQ studies (e.g., <cit.>). Nevertheless, adaptations are needed to handle other common types of joint modeling, such as hierarchical or graph-based approaches.







§ ESTIMATION PROCEDURE FOR A GIVEN QOI
 
 


 §.§ Quantile estimation procedure


For all forward models G, a QoI of the output Y often cannot be computed explicitly. Nevertheless, an estimate can be provided by using a statistical estimation procedure. We describe here how we construct this estimator, following  the importance sampling approach suggested by <cit.>. For better reading and to fit our experiments, we consider that all inputs X=(X_1,...,X_d)
are independent, but generalizing the approach to non-independent inputs is straightforward. 

By the law of large numbers, the estimator F_iδ (t) converges almost surely to the cumulative distribution function of Y^θ for all t∈ℝ. The quantile estimator q̂_iδ of q^α(Y^iδ) is built by plug-in:

    q̂_iδ := inf{ t ∈ℝ |  F_iδ (t) ≥α}.

Hence q^α(Y^iδ) can be estimated for all i and for all δ given only a sample of P_X. The estimator of the PLI robustness index S_i (Equation (<ref>)) is then given by

    S_i(f_iδ) := q̂_iδ/q̂_i - 1,

where q̂_i is the empirical quantile estimator of q^α(Y) where Y:=G(X) and X∼ f_1⊗···⊗ f_d. The asymptotic properties of S_i are studied in <cit.>.

The estimators F_iδ(t) and q̂_iδ can be seen as functions of the parameter θ∈Θ_i. The space Θ_i is a parameter space of the family 𝒫_i of possible perturbation densities of f_i. Since the estimator F_iδ(t) is piecewise constant as a function of t, jumping on the sample points Y_j, and as a function of θ, its generalized inverse q̂_iδ is also piecewise constant taking values in {Y_1,...,Y_N} 

    θ∈Θ_i →q̂_iδ∈{Y_1,...,Y_N}.

Similarly since q̂_i also takes finite values, this shows that the PLI estimator S_i also takes finite values and 

    min_f_iδ∈Λ_iδS_i(f_iδ)     and   max_f_iδ∈Λ_iδS_i(f_iδ)

is a discrete optimization problem. Therefore, a gradient-based optimization method cannot be used directly.



 §.§ Bootstrap confidence intervals

We used a bootstrap procedure to compute confidence intervals for

    min_f_iδ∈Λ_iδ S_i(f_iδ)     and   max_f_iδ∈Λ_iδ S_i(f_iδ).

Focusing only on the maximization problem, for each δ, we approximated the maximum value of S_i on Λ_iδ by
max_f_iδ∈Λ̅_iδS_i(f_iδ),
where Λ̅_iδ is a discretization of the FR sphere Λ_iδ (100 points in our simulation). Next we compute the maximum points f_iδ^* on the discrete FR sphere Λ_iδ directly. This means that 
S_i (f_iδ^*) = max_f_iδ∈Λ̅_iδS_iδ,
and since S_i (f_iδ^*) is an estimator built on the sample {Y_1,...,Y_N}, we can use a bootstrap procedure to obtain a confidence interval for max_f_iδ∈Λ̅_iδS_i (f_iδ). This is expected to give a confidence interval for 
max_f_iδ∈Λ_iδS_i (f_iδ),
provided Λ̅_iδ is refined enough.



§ MORE DETAILS ON GEOMETRY AND PROOFS



 §.§ Computation of Fisher-Rao spheres using geodesics
 

As explained in Section <ref>, the sphere approximation algorithm is based on Proposition <ref>. But this method may fail for large radii. The reasons of this failure may be either the incompleteness of the manifold or finiteness of the injectivity radius, which will be defined in the following.  



  §.§.§ Completeness

A connected Riemannian manifold (M,g) is geodesically complete if all geodesics γ are defined for all time t∈ℝ. This is equivalent to the completeness of M as a metric space endowed with the geodesic distance d, by the Hopf-Rinow theorem <cit.>. This theorem additionally states that a connected manifold M is geodesically complete if and only if every closed and bounded set in M is compact. This implies that if the manifold is not complete, then there may exist some spheres (which are always closed and bounded) that are not compact. Note that, around a given point, spheres with small enough radii are always compact. Therefore, the non-compactness of spheres may only arise for large enough radii.

For the sphere approximation algorithm from Section <ref>, approximating non-compact spheres can lead to numerical issues. This is because for a given chart on which the geodesic equation is numerically solved, depending on the initial conditions, the solution (the geodesic) γ(t) may blow-up in finite time t_*. Hence, it will be harder for the approximated solution γ̅(t) to be accurate as t gets closer to t_*.
 


  §.§.§ Injectivity radius

Let (M,g) be a connected Riemannian manifold. If the injectivity radius ι_M=∞, then Proposition <ref> can be used for all δ>0. On the contrary, if ι_M<∞, the radius δ has to be small enough or the manifold needs to verify some additional properties in order for Proposition <ref> to hold for all δ>0. More precisely, let us first define the exponential map at p∈ M

    exp_p  :     Ω_p ⊂ T_p M ⟶ M 
        v ⟼γ_p,v(1).

Here, γ_p,v is the unique geodesic verifying the conditions γ_p,v(0)=p and γ̇_p,v(0)=v. The set Ω_p is defined as the set of vectors v ∈ T_p M such that γ_p,v is defined until time t=1. It is a neighborhood of the null vector 0_p in T_p M. In addition, the exponential map exp_p is a local diffeomorphism in a small neighborhood around 0_p. 
The question whether exp_p is a global diffeomorphism is adressed using the notion of injectivity radius. 

Let us first denote B(p,r) and 𝒮(p,r) resp. the open ball and sphere in M centered at p with radius r for the geodesic distance. Further, let us denote b(0_p,r):= {v ∈ T_p M  |  |v|_p< r} and s(0_p,r):= {v ∈ T_p M  |  |v|_p=r} resp. the open ball and sphere centered at 0_p with radius r in T_p M. Since exp_p is a local diffeomorphism around 0_p for all p in M, there exists r_p>0 such that for r≤ r_p we have exp_p : b(0_p,r_p) → B(p,r_p) is a diffeomorphism. We also have that exp_p(s(0_p,r_p)) = 𝒮(p,r_p) and exp_p(b(0_p,r_p)) = B(p,r_p). The injectivity radius ι_p at point p is defined as the largest radius r for which exp_p is a diffeomorphism onto the geodesic ball B(p,r)⊂ M. That is,

    ι_p := sup{ r > 0  |  exp_p  is a diffeomorphism on  b(0_p,r) }.

Further, the injectivity radius of the manifold is defined as ι_M:= inf_p∈ Mι_p. 

If ι_M is infinite, then exp_p : T_p M → M is a global diffeomorphism (for any p in M). This implies that for all r>0 the spheres 𝒮(p,r) in M can be identified with

    𝒮(p,r)=exp_p(s(0_p,r))={γ(1)  |  γ geodesic s.t. γ(0)=p, γ̇(0)=v  and  |v|_p=δ}.

Hence, this justifies the use of the initial algorithm from Section <ref> for all radius r>0. This also implies that the manifold is complete.

Now, if the injectivity radius ι_M<∞, then there exists p ∈ M such that for r≥ι_p the exponential map exp_p fails to be a diffeomorphism from b(0_p,r) to B(p,r). 
The good news is that if the manifold is complete, the exponential map is always onto 𝒮(p,r) 

    𝒮(p,r) ⊂exp_p( s(0_p,r)).

This guaranties that the sphere approximation algorithm from Section <ref> gives at least some points that approximate 𝒮(p,r) (but also some that do not necessarily fall on 𝒮(p,r)). This is true since M is assumed to be connected.
In general, it is not obvious to determine whether ι_M is infinite or not. Some curvature conditions (Hadamard theorem, see <cit.>) lead to lower bounds for ι_p. Nevertheless, in practice these conditions are hard to verify. 



 §.§ Proofs
 


The fact that ℛ_B and 𝒫_A have the same FIM is a direct consequence of the first assertion and the standard pushforward theorem. Indeed, if we assume ℛ_B=h_#𝒫_A, then it implies that they have the same FIM. Moreover, since 𝒫_A and h_#𝒫_A have the same FIM (standard pushforward theorem), the second assertion is obvious.

Now let us prove the first assertion. Let C be a measurable subset on 𝒴. We have on the one hand

    h_#(P_θ^A)(C) = P_θ^A(h^-1(C)) = P_θ(h^-1(C)∩ A)/P_θ(A).
 
On the other hand, 
we have

    (h_#P_θ)^B(C) = h_#P_θ(C∩ B)/h_#P_θ(B)=P_θ(h^-1(C∩ B))/P_θ(h^-1(B))   =P_θ(h^-1(C)∩ h^-1(B))/P_θ(A)
       =P_θ(h^-1(C)∩ A)/P_θ(A).

This allows to conclude the proof. Note that h does not need be bijective in the proof.



Under the previous notations, if we now assume that h:𝒳→𝒴 is a sufficient statistic for 𝒫, then the first assertion of Lemma <ref> still holds. Note that in this case, h is also a sufficient statistic of a truncated version of 𝒫 i.e. 𝒫_A (see <cit.>). In addition, the Fisher-Neyman factorisation theorem (see <cit.>) implies that

    p_θ(x) = u(x) r_θ(h(x)),

where p_θ and r_θ are resp. the pdf of P_θ and R_θ:=h_# P_θ and u is a non-negative function on 𝒳. Let us denote resp. p_θ^A and r_θ^B the truncated pdfs of p_θ on A and r_θ on B. We can compute the FIM of K_θ^B of the truncated family ℛ_B={R_θ^B}_θ∈Θ,

    (K_θ^B)_ij   = ∫_𝒴(∂_i log r_θ^B(y))(∂_j log r_θ^B(y)) dR_θ^B(y) 
       = ∫_𝒳(∂_i log r_θ^B(h(x)))(∂_j log r_θ^B(h(x))) dP_θ^A(x) 
       = ∫_𝒳(∂_i logp_θ(x)/u(h(x))R_θ(B))(∂_j logp_θ(x)/u(h(x))R_θ(B)) dP_θ^A(x)  
       = ∫_𝒳(∂_i logp_θ(x)/P_θ(A))(∂_j logp_θ(x)/P_θ(A)) dP_θ^A(x) 
       = ∫_𝒳(∂_i log p_θ^A(x))(∂_j log p_θ^A(x)) dP_θ^A(x) 
       = (I_θ^A)_ij,

where (B1) is obtained from the transfer theorem, (B2) and (B4) are resp. obtained by the definition of r_θ^B and p_θ^A as well as the Fisher-Neyman theorem and (B3) holds because u(h(x)) does not depend on θ and R_θ(B)=P_θ(A).    



    Let p_θ:ℝ→ (0,∞) be defined by 

    p_θ(x) = 1/s p(x-m/s),

where θ=(m,s) ∈ℝ× (0,∞) is the parameter. Here, p is the initial density. Let us compute the FIM for this family. First, taking the partial derivatives of log p_θ we get

    ∂_m log p_θ(x) = -1/s p'( x-m/s)/p( x-m/s)      and      ∂_s log p_θ(x) =( -x-m/s^2) p'( x-m/s)/p( x-m/s)  -1/s.


Now, we can compute 

    (I_θ)_11 = ∫_ℝ(-1/s p'( x-m/s)/p( x-m/s))^2 1/sp( x-m/s) dx
        = 1/s^2α
 

    (I_θ)_12   = ∫_ℝ(-1/s p'( x-m/s)/p( x-m/s)) ( ( -x-m/s^2) p'( x-m/s)/p( x-m/s)  -1/s) 1/sp( x-m/s) dx= 1/s^2γ


    (I_θ)_22 = ∫_ℝ(( -y/s) p'(y)/p(y)  -1/s)^2 1/sp(y) s dy
        =1/s^2β.

Here α,β,γ are given by

    α=∫_ℝ( p'(y)/p(y))^2 p(y) dy,    β=∫_ℝ(yp'(y)/p(y)  + 1)^2 p(y) dy

and 

    γ = ∫_ℝ(-p'(y)/p(y)) (  -yp'(y)/p(y) - 1 ) p(y) dy.

We refer to <cit.> for a similar computation in higher dimension. We assume that these quantities are well defined and finite. In this case, the FIM for the location-scale family of p is given by 

    I_θ = 1/s^2[ α γ; γ β ].

The matrix [ α γ; γ β ] can be diagonalized on an orthonormal basis of eigenvectors

    I_θ =  Q_αβγ(1/s^2[ λ_1   0;   0 λ_2 ]) Q_αβγ^⊺ = Q_αβγ[ √(λ_1)      0;      0 √(λ_2) ](1/s^2[ 1 0; 0 1 ])
        [ √(λ_1)      0;      0 √(λ_2) ]Q_αβγ^⊺.
 
Here, Q_αβγ is an orthogonal change-of-basis matrix and λ_1,λ_2≥0 are the eigenvalues and they depend on α,β,γ. By denoting P_αβγ := [ √(λ_1)      0;      0 √(λ_2) ]Q_αβγ we thus get 

    I_θ = P_αβγ(1/s^2[ 1 0; 0 1 ]) P_αβγ^⊺.

This proves that the FIM is, up to reparameterization, the Poincaré metric on ℍ := {(x,y) ∈ℝ^2  |  y >0}.



For any location-scale family, the change-of-basis matrix P_αβγ depends on α,β,γ therefore having a good approximation of these integral quantities will allow us to approximate P_αβγ. This will in turn allow us to compute geometric quantities (distance, curvature, angle,...) for the location-scale family using only the Poincaré metric. Indeed, these quantities are already known for the latter.
    


 §.§ Explicit Fisher-Rao distance computation in the triangular and exponential distributions
 
The triangular and exponential distribution families only depend on a single parameter. We can thus easily compute the Fisher information and the Fisher-Rao distance exlicitly.


  §.§.§ Triangular family

The triangular family 𝒯_[a,b] of densities q_m on the interval [a,b] is given by

    q_m(x)= 2(x-a)/(b-a)(m-a)1_x∈ [a,m] + 2(b-x)/(b-a)(b-m)1_x∈ (m,b],

where m∈(a,b) is the parameter. The Fisher information i_m is computed as follows: 
    
  * for x∈ [a,m], 
    ∂_m log q_m(x) =  - ∂_m log(m-a) = -1/m-a,
    
  * for x ∈ (m,b],
    ∂_m log q_m(x) = -∂_m log(b-m) = 1/b-m.

Hence, we have

    i_m    = ∫_a^b (∂_m log q_m(x))^2 q_m(x) dx 
       = ∫_a^m 1/(m-a)^22(x-a)/(b-a)(m-a) dx + ∫_m^b 1/(b-m)^22(b-x)/(b-a)(b-m) dx 
       = 1/(m-a)^3(b-a)(m-a)^2 + 1/(b-m)^3(b-a)(b-m)^2
       = 1/b-a(1/m-a + 1/b-m) = 1/(m-a)(b-m).

As {q_m : m ∈ (a,b)} is a one parameter family, every geodesic correspond to a segment γ(t) = m_0 + tv, up to reparameterization. But, since the integral in the definition of the Fisher-Rao distance 
is invariant under reparameterization, we can explicitly compute the distance between two points using the segment γ(t) = m_0 + tv. To begin with, let α:= (a+b)/2 and β := (a-b/2)^2. We have (b-m)(m-a) = -(m-α)^2 + β. Then, for m_0,m_1 ∈ (a,b), we compute 

    d(m_0,m_1)    =| ∫_m_0^m_1dm/√(-(m-α)^2 + β)| = |∫_m_0^m_1dm/√(β)√(-(m-α/√(β))^2 + 1)|  
       u:=m-α/√(β)=|∫_m_0-α/√(β)^m_1-α/√(β)du/√(1-u^2)| 
       =| arcsin(m_1-α/√(β))-arcsin(m_0-α/√(β))|.

This means that the Fisher-Rao sphere centered at m_0 ∈ (a,b) with small radius δ>0, is explicitly determined by the following two points

    m_+ = √(β)sin(δ + arcsin(m_0-α/√(β)) ) + α,    m≥ m_0, 
    
    m_- = √(β)sin(arcsin(m_0-α/√(β)) - δ) + α,    m < m_0.





  §.§.§ Exponential distributions family


The exponential family ℰ has densities given by

    p_λ(x) = λ e^-λ x1_x>0,    λ∈ (0,∞).

The Fisher information is given by i_λ = λ^-2 (see <cit.>, page 25). Similar to the triangular family, the Fisher-Rao geodesic distance in this case is

    d(λ_0,λ_1) = | ∫_λ_0^λ_1√(1/λ^2) dλ| =  |log(λ_1/λ_0) |,    λ_0,λ_1>0.

So that, the Fisher-Rao sphere centered at λ_0 with radius δ is {λ_-,λ_+} given by

    λ_+ = λ_0 e^δ,    λ_+≥λ_0,
    λ_- = λ_0 e^-δ,    λ_-≤λ_0.

Note that this family has infinite diameter since 

    diam(ℰ) := sup_λ_0,λ_1 ∈ (0,∞) d(λ_0,λ_1)=sup_λ_0,λ_1 ∈ (0,∞)|log(λ_1/λ_0) |=∞.







§ DUALISTIC STRUCTURE IN INFORMATION GEOMETRY
 
As explained in the main article, the Riemannian structure of 𝒫 allows to obtain the Levi-Civita affine connection ∇̅. In addition, Eguchi's formulas <cit.> endows the space 𝒫 with an additional dualistic structure:

    Γ_ij,k (θ_0) =  - ∂_θ^i∂_θ^j∂_θ_0^k[ 𝒟(P_θ | P_θ_0) ]_θ=θ_0,
    
        ^*Γ_ij,k (θ_0) =  - ∂_θ^k∂_θ_0^i∂_θ_0^j[ 𝒟(P_θ | P_θ_0) ]_θ=θ_0.

The coefficients Γ_ij,k and ^*Γ_ij,k are the so-called Christoffel symbols (of the first kind) of two affine connections on 𝒫 denoted by ∇ and ∇^* that verify a dualistic (conjugacy) property <cit.>. This property implies that the unique Levi-Civita connection ∇̅ (resp. Christoffel symbols Γ̅) of the Fisher metric g is given by the average of the two dual connections (resp. Christoffel symbols)

    ∇̅= ∇ +  ^*∇/2   and   Γ̅_ij,k = Γ_ij,k +  ^*Γ_ij,k/2.

The Chistoffel symbols of the first kind Γ̅_ij,k and the Christoffel symbols (of the second kind) Γ̅_ij^k defined in the main article are related by the following formula

    Γ̅_ij^k = ∑_l=1^m Γ̅_ij,l g^lk.

Here, g^lk is the inverse matrix coefficients of g. These two previous formulas provide another way for computing the Levi-Civita Christoffel symbols. This will be useful in Section <ref> for the truncated normal family.


 


§ COMPUTATIONS OF FIM AND CHRISTOFFEL SYMBOL COEFFICIENTS
 

We will now compute the FIM coefficients and the Christoffel symbols for some families of probability distributions. This will prove Proposition 3.1 from the main article. These computations are motivated by the fact that we need to compute the Christoffel symbols in order to approximate the Fisher-Rao spheres.



 §.§ Truncated normal family
 
We will explicitly compute the FIM I_θ^[a,b] and Christoffel symbols Γ̅_ij,k of the truncated normal family 𝒩_[a,b]. We use the formula <cit.>

    (I_θ^[a,b])_ij = -∂_θ^i∂_θ_0^j[𝒟(q_θ | q_θ_0)]_θ=θ_0.

Here, I_θ^[a,b] is the FIM and q_θ is the truncated density and 𝒟 is the Kullback-Leibler divergence defined as 𝒟(q_θ| q_θ_0):= 𝔼_X∼ P_θ[log(q_θ/q_θ_0(X))]. For the Christoffel symbols Γ̅_ij,k, we first compute Γ_ij,k and Γ_ij,k^* (see Section <ref> for explanation)

    Γ_ij,k (θ_0) =  - ∂_θ^i∂_θ^j∂_θ_0^k[ 𝒟(q_θ | q_θ_0)]_θ=θ_0   and   ^* Γ_ij,k  (θ_0) =  - ∂_θ^k∂_θ_0^i∂_θ_0^j[ 𝒟(q_θ | q_θ_0) ]_θ=θ_0.

Then, we take the average to get the Levi-Civita Christoffel symbols Γ̅_ij,k.



  §.§.§ Computing the conditional expectation and variance of Gaussian variables

Let B=[a,b], it denotes the truncation domain. Denoting P_θ =𝒩(μ,σ^2), for θ=(μ,σ), with density p_θ, the conditional expectation and variance over B of a Gaussian random variable is given by 

    μ_B := 𝔼_X∼ q_θ(X) =  1/P_θ(B)∫_B x p_θ(x) dx = -σ^2/P_θ(B)∫_B -(x -μ)/σ^2 p_θ(x) dx + μ = -σ^2 [q_θ]_a ^b + μ,

 and

    σ_B^2     := Var_X∼ q_θ(X) 
       =1/P_θ(B)∫_B (x-μ_B)^2p_θ(x) dx 
       = 1/P_θ(B)( ∫_B (x-μ)^2p_θ(x) dx  + 2(μ - μ_B)∫_B (x-μ) p_θ(x) dx + ∫_B (μ-μ_B)^2p_θ(x) dx  ) 
       = -σ^2/P_θ(B)([ (x-μ)p_θ]_a^b -∫_B p_θ(x)dx   ) -2(μ - μ_B)^2 + (μ-μ_B)^2
       = -σ^2 ( [ (x-μ)q_θ]_a^b -1 ) - (μ - μ_B)^2.





  §.§.§ Computing the Kullback-Leibler divergence of truncated Gaussian densities

Let us compute the Kullback-Leibler divergence on the truncated normal family 

    D(θ,θ_0) := 𝒟( q_θ | q_θ_0) = ∫ q_θlog(q_θ/q_θ_0) = ∫ q_θlog q_θ - ∫ q_θlog q_θ_0.

Setting A_θ := ∫ q_θlog q_θ, we have

    D(θ,θ_0)    = A_θ - ∫ q_θlog q_θ_0
       = A_θ - ∫_B p_θ/P_θ(B)log p_θ_0 + ∫_B p_θ/P_θ(B)log P_θ_0(B) 
        = A_θ - ∫_B p_θ/P_θ(B)log p_θ_0 + log P_θ_0(B)_=:C_θ_0
        = A_θ + C_θ_0 - 1/P_θ(B)∫_B p_θlog p_θ_0 ,

A_θ and C_θ_0 depend on θ and θ_0 respectively. Let us now compute the last term explicitly:

    ∫_B p_θlog p_θ_0   = ∫_B p_θ( -log (√(2π)σ_0) - (x-μ_0)^2/2σ_0^2) 
        = -log (√(2π)σ_0) P_θ(B) -1/2σ_0^2∫_B (x-μ_0)^2 p_θ.

We finally obtain

    1/2σ_0^2∫_B (x-μ_0)^2 p_θ   = 1/2σ_0^2(∫_B (x-μ_B)^2 p_θ + 2(μ_B - μ_0) ∫_B (x-μ_B)p_θ_=0 +(μ_B - μ_0)^2 P_θ(B) )
        = P_θ(B) σ_B^2/2σ_0^2 +(μ_B - μ_0)^2/2σ_0^2 P_θ(B).


Now we can express (<ref>) using μ_B and σ_B^2 (which depend on θ)

    D(θ,θ_0) = A_θ + C_θ_0  + log(√(2π)) + logσ_0 + σ_B^2 + (μ_B - μ_0)^2/2σ_0^2.




  §.§.§ Computing the Fisher Information matrix
 

In the computation of the FIM using (<ref>), we only end up with terms that depend both on θ_0 and θ. This means that A_θ and C_θ_0 disappear. Therefore, we omit them from the start. For ∂_μ_0 D and ∂_σ_0 D we obtain 

    ∂_μ_0 D = 2(-1)(μ_B - μ_0)/2σ_0^2= -μ_B - μ_0/σ_0^2    and    ∂_σ_0 D = 1/σ_0 - σ_B^2 + (μ_B -μ_0)^2/σ_0^3.

Now for ∂_μ∂_μ_0 D, ∂_σ∂_μ_0 D and ∂_σ∂_σ_0 D, we get

    ∂_μ∂_μ_0 D     = - ∂_μμ_B(θ) /σ_0^2,
    ∂_σ∂_μ_0 D    = -∂_σμ_B(θ)/σ_0^2,
    ∂_σ∂_σ_0 D    = -1/σ_0^3( ∂_σσ_B^2(θ) + 2(μ_B - μ_0) ∂_σμ_B(θ) ).
 

Let us now compute the terms ∂_μμ_B(θ), ∂_σμ_B (θ) and ∂_σσ_B^2 (θ):

    ∂_μμ_B(θ)    = 1 - σ^2 [∂_μ q_θ(x) ]_a^b = 1 - σ^2 [ q_θ(x)( x-μ/σ^2 + [q_θ]_a^b) ]_a^b,
    ∂_σμ_B(θ)    = -2σ[ q_θ(x) ]_a^b -σ^2 [ ∂_σ q_θ(x) ]_a^b
       = - 2 σ[q_θ]_a^b  -σ[ q_θ(x)([(x- μ)q_θ]_a^b - 1  +(x-μ)^2/σ^2) ]_a^b
       = -σ[ q_θ(x)([(x- μ)q_θ]_a^b + 1  +(x-μ)^2/σ^2) ]_a^b
    ∂_σ
    σ_B^2(θ)    = 2σ - 2σ[ (x-μ)q_θ]_a^b - σ^2[ (x-μ)∂_σ q_θ]_a^b - 2(μ-μ_B)∂_σμ_B(θ) 
       = 2σ( 1 - [ (x-μ)q_θ]_a^b ) - σ^2[ (x-μ)q_θ(x)( x-μ/σ^2 + [q_θ]_a^b) ]_a^b
        - 2(μ-μ_B)∂_σμ_B(θ).


Lastly, coming back to (<ref>), (<ref>) and (<ref>) we get

    ∂_μ∂_μ_0 D     = - ∂_μμ_B(θ) /σ_0^2 = - 1/σ_0^2 + σ^2 [ q_θ(x)( x-μ/σ^2 + [q_θ]_a^b) ]_a^b/σ_0^2,
    ∂_σ∂_μ_0 D    = -∂_σμ_B(θ)/σ_0^2 = σ[ q_θ(x)([(x- μ)q_θ]_a^b + 1  +(x-μ)^2/σ^2) ]_a^b/σ_0^2,
    ∂_σ∂_σ_0 D    = -1/σ_0^3( ∂_σσ_B^2(θ) + 2(μ_B - μ_0) ∂_σμ_B(θ) ) 
       = -1/σ_0^3(2σ( 1 - [ (x-μ)q_θ]_a^b ) - σ^2[ (x-μ)q_θ(x)( x-μ/σ^2 + [q_θ]_a^b) ]_a^b  ).
 
Further, taking θ=θ_0 and multiplying by -1, we obtain

    (I_θ_0)_11=-∂_μ∂_μ_0 D_|θ=θ_0   =  1/σ_0^2 - [ q_θ_0(x)( x-μ_0/σ_0^2 + [q_θ_0]_a^b) ]_a^b,
    
        (I_θ_0)_12 =-∂_σ∂_μ_0 D_|θ=θ_0   = -σ_0 [ q_θ_0(x)([(x- μ_0)q_θ_0]_a^b + 1  +(x-μ_0)^2/σ_0^2) ]_a^b/σ_0^2,
    
    (I_θ_0)_22=-∂_σ∂_σ_0 D_|θ=θ_0   = 2/σ_0^2 - 2 [ (x-μ_0)q_θ_0]_a^b /σ_0^2 -  [ (x-μ_0)/σ_0 q_θ_0(x)( x-μ_0/σ_0^2 + [q_θ_0]_a^b) ]_a^b  .
 


  §.§.§ A few remarks on the truncated normal family

Here, we observe that when a→ -∞ and b→∞, we get

    (I_θ_0)_11→1/σ_0^2,     (I_θ_0)_12→ 0    and     (I_θ_0)_22→2/σ_0^2.


In other words, I_θ^[a,b] converges to the FIM of the non-truncated normal family I_θ. 

Now, if we fix a and b, for θ=(μ,σ) such that μ∈ (a,b) and σ is close to 0, then the normalization constant N_θ:= ∫_a^b (√(2π)σ)^-1exp(-(x-μ)^2/(2σ^2) dx
is close to 1. Indeed, most of the mass of the normal density p_θ is already inside [a,b]. As a consequence, q_θ and 𝒩(μ,σ^2) are close. So that, for such θ, the matrices I_θ and I_θ^[a,b] are also close. This means that in areas where μ∈ (a,b) and σ is close to 0, both of these metrics induce a very similar geometry.













  §.§.§ Computing the Levi-Civita Christoffel symbols

As explained previously, the Levi-Civita Christoffel symbols Γ̅_ij,k is computed by averaging the Christoffel symbols of Γ_ij,k and ^*Γ_ij,k

    Γ̅_ij,k = Γ_ij,k +  ^*Γ_ij,k/2.

It remains to express it as a function of μ_B and σ_B^2 and of their partial derivatives w.r.t. μ and σ.


  
Computing ^* Γ_ij,k 

First, for ∂_μ_0 D and ∂_σ_0 D we have

    ∂_μ_0 D    = -μ_B - μ_0/σ_0^2,
    ∂_σ_0 D    = 1/σ_0 - σ_B^2 + (μ_B -μ_0)^2/σ_0^3.
 

We differentiate these two expressions with respect to ∂_μ_0 and ∂_σ_0 to obtain ∂_μ_0∂_μ_0 D, ∂_μ_0∂_σ_0 D and ∂_σ_0∂_σ_0 D :

    ∂_μ_0∂_μ_0 D    = 1/σ_0^2,
    ∂_μ_0∂_σ_0 D    = 2(μ_B - μ_0)/σ_0^3,
    ∂_σ_0∂_σ_0 D    = -1/σ_0^2 + 3σ_B^2 + (μ_B - μ_0)^2/σ_0^4.


Now, it remains to differentiate these three expressions w.r.t. to μ and σ to obtain the following six functions ∂_μ∂_μ_0∂_μ_0 D, ∂_σ∂_μ_0∂_μ_0 D,  ∂_μ∂_μ_0∂_σ_0 D, ∂_σ∂_μ_0∂_σ_0 D, ∂_μ∂_σ_0∂_σ_0 D and ∂_σ∂_σ_0∂_σ_0 D. We obtain 

    ∂_μ∂_μ_0∂_μ_0 D    = 0,
    ∂_σ∂_μ_0∂_μ_0 D    =  0,
    ∂_σ∂_μ_0∂_σ_0 D    =  2 ∂_σ [μ_B(θ) ]/σ_0^3,
    ∂_μ∂_σ_0∂_σ_0 D    = 3/σ_0^4( ∂_μ [σ_B^2(θ) ] + 2 ∂_μ [μ_B(θ)] (μ_B-μ_0) ),
    ∂_σ∂_σ_0∂_σ_0 D    = 3/σ_0^4( ∂_σ [σ_B^2(θ) ] + 2 ∂_σ [μ_B(θ)] (μ_B-μ_0) ).


Let us now compute the partial derivatives of μ_B and σ_B^2 in terms of the truncated Gaussian density q_θ
 
    ∂_μμ_B(θ)    = 1 - σ^2 [ ∂_μ q_θ]_a^b,
    ∂_σμ_B(θ)    = -2σ[ q_θ]_a^b - σ^2 [ ∂_σ q_θ]_a^b,
    ∂_μσ_B^2 (θ)    = - σ^2  [- q_θ +(x-μ) ∂_μ q_θ]_a^b - 2 (μ - μ_B) ( 1 - ∂_μ[μ_B(θ)] ),
    ∂_σσ_B^2(θ)    = 2σ( 1 - [ (x-μ)q_θ]_a^b ) - σ^2[ (x-μ)∂_σ q_θ]_a^b  + 2(μ-μ_B)∂_σμ_B(θ).

    
    
    

Coming back to the symbols ^* Γ_ij,k, we have

    ^* Γ_11,1   =  -∂_μ∂_μ_0∂_μ_0 D_|θ = θ_0 = 0, 
    
    ^* Γ_11,2   = -∂_σ∂_μ_0∂_μ_0 D_|θ = θ_0 =  0,
    
    ^* Γ_12,1   =  ^*Γ_21,1= -∂_μ∂_μ_0∂_σ_0 D_|θ = θ_0 = - 2 ∂_μ [μ_B(θ_0) ]/σ_0^3,
    
    ^* Γ_12,2   = ^* Γ_21,2= -∂_σ∂_μ_0∂_σ_0 D_|θ = θ_0 =  -2 ∂_σ [μ_B(θ_0) ]/σ_0^3,
    
    ^* Γ_22,1   = -∂_μ∂_σ_0∂_σ_0 D_|θ = θ_0 = -3/σ_0^4( ∂_μ [σ_B^2(θ_0) ] + 2 ∂_μ [μ_B(θ_0)] (μ_B(θ_0)-μ_0) ),
    
    ^* Γ_22,2   = -∂_σ∂_σ_0∂_σ_0 D_|θ = θ_0 = -3/σ_0^4( ∂_σ [σ_B^2(θ_0) ] + 2 ∂_σ [μ_B(θ_0)] (μ_B(θ_0)-μ_0) ).

   







  
Computing Γ_ij,k

We have already computed  ∂_μ_0 D and ∂_σ_0 D 

    ∂_μ_0 D = -μ_B - μ_0/σ_0^2   and   ∂_σ_0 D = 1/σ_0 - σ_B^2 + (μ_B -μ_0)^2/σ_0^3.

We also computed ∂_μ∂_μ_0 D, ∂_σ∂_μ_0 D and ∂_σ∂_σ_0 D

    ∂_μ∂_μ_0 D     = - ∂_μμ_B(θ) /σ_0^2,
    ∂_σ∂_μ_0 D    = -∂_σμ_B(θ)/σ_0^2,
    ∂_σ∂_σ_0 D    = -1/σ_0^3( ∂_σσ_B^2(θ) + 2(μ_B - μ_0) ∂_σμ_B(θ) ).
 
It remains to compute ∂_μ∂_σ_0 D

    ∂_μ∂_σ_0  D = -1/σ_0^3( ∂_μ [σ_B^2]  + 2 ∂_μ[μ_B](μ_B - μ_0) ).

Now differentiating the last four expressions with respect to ∂_μ and ∂_σ, we obtain

    ∂_μ∂_μ∂_μ_0 D     = - ∂_μ^2 μ_B(θ) /σ_0^2,
    ∂_σ∂_μ∂_μ_0 D     = - ∂_σμ ^2 μ_B(θ) /σ_0^2,
    ∂_σ∂_σ∂_μ_0 D    = -∂_σ^2 μ_B(θ)/σ_0^2,
    ∂_μ∂_μ∂_σ_0 D    = -1/σ_0^3( ∂_μ^2 σ_B^2(θ) + 2(μ_B - μ_0) ∂_μ^2 μ_B(θ) + 2(∂_μμ_B(θ) )^2 ),
    ∂_μ∂_σ∂_σ_0 D    = -1/σ_0^3( ∂_μ∂_σσ_B^2(θ) + 2(μ_B - μ_0) ∂_μσ^2 μ_B(θ) + 2(∂_σμ_B(θ) )(∂_μμ_B(θ) ) ),
    ∂_σ∂_σ∂_σ_0 D    = -1/σ_0^3( ∂_σ^2 σ_B^2(θ) + 2(μ_B - μ_0) ∂_σ^2 μ_B(θ) + 2(∂_σμ_B(θ) )^2 ).
 
Thus, the Christoffel symbols Γ_ij,k are given by

    Γ_11,1   = -∂_μ∂_μ∂_μ_0 D_|θ=θ_0  = ∂_μ^2 μ_B(θ_0) /σ_0^2,
    Γ_21,1   =Γ_12,1 = -∂_σ∂_μ∂_μ_0 D_|θ=θ_0  =  ∂_σμ ^2 μ_B(θ_0) /σ_0^2,
    Γ_22,1   = -∂_σ∂_σ∂_μ_0 D_|θ=θ_0 = ∂_σ^2 μ_B(θ_0)/σ_0^2,
    Γ_11,2   = -∂_μ∂_μ∂_σ_0 D_|θ=θ_0 = 1/σ_0^3( ∂_μ^2 σ_B^2(θ_0) + 2(μ_B(θ_0) - μ_0) ∂_μ^2 μ_B(θ_0) + 2(∂_μμ_B(θ_0) )^2 ),
    Γ_12,2   =Γ_21,2 = -∂_μ∂_σ∂_σ_0 D_|θ=θ_0
       = 1/σ_0^3( ∂_μ∂_σσ_B^2(θ_0) + 2(μ_B(θ_0) - μ_0) ∂_μσ^2 μ_B(θ_0) +2(∂_σμ_B(θ_0) )(∂_μμ_B(θ_0) ) ),
    Γ_22,2   = -∂_σ∂_σ∂_σ_0 D_|θ=θ_0 = 1/σ_0^3( ∂_σ^2 σ_B^2(θ_0) + 2(μ_B(θ_0) - μ_0) ∂_σ^2 μ_B(θ_0) + 2(∂_σμ_B(θ_0) )^2 ).
 



  
First and second order partial derivatives of μ_B and σ_B^2 with respect to (μ,σ)
The two conjugate Christoffel symbols Γ_ij,k and Γ_ij,k^* are functions of the first and second order partial derivatives of μ_B and σ_B^2. Now we explicitly compute these derivatives. To begin with, let us compute the first order partial derivatives of μ_B:

    ∂_μμ_B(θ)    = 1 - σ^2 [ ∂_μ q_θ]_a^b,
    ∂_σμ_B(θ)    = -2σ[ q_θ]_a^b - σ^2 [ ∂_σ q_θ]_a^b,
    ∂_μσ_B^2 (θ)    = - σ^2  [- q_θ +(x-μ) ∂_μ q_θ]_a^b - 2 (μ - μ_B) ( 1 - ∂_μ[μ_B(θ)] ),
    ∂_σσ_B^2(θ)    = 2σ( 1 - [ (x-μ)q_θ]_a^b ) - σ^2[ (x-μ)∂_σ q_θ]_a^b + 2(μ-μ_B)∂_σμ_B(θ).

For the second order partial derivatives of μ_B and σ_B ^2, we obtain 

    ∂_μ ^2 μ_B(θ)    = -σ^2 [∂_μ ^2 q_θ]_a ^b,
    ∂_σμ ^2 μ_B(θ)    = -2 σ[ ∂_μ q_θ]_a ^b - σ^2 [ ∂_σμ ^2 q_θ]_a ^b,
    ∂_σ ^2 μ_B(θ)    = -2[q_θ]_a ^b -4σ[∂_σ q_θ]_a ^b - σ^2 [∂_σμ ^2 q_θ]_a ^b,


    ∂_μ^2 σ_B^2    =-σ^2 [-2∂_μ q_θ + (x-μ) ∂_μ ^2 q_θ]_a^b -2(1-∂_μμ_B(θ) )^2 + 2(μ - μ_B) ∂_μ ^2 μ_B, 
    ∂_σμ^2 σ_B^2    =  -2σ[-q_θ + (x-μ) ∂_μ q_θ]_a^b - σ^2 [ -∂_σ q_θ + (x-μ)∂_σμ ^2 q_θ]_a^b
       + 2 ∂_σμ_B (1- ∂_μμ_B) + 2∂_σμ ^2  μ_B (μ - μ_B),
    ∂_σ^2 σ_B^2    =  2(1-[(x-μ) q_θ]_a^b )- 4σ[(x-μ) ∂_σ q_θ]_a^b - σ^2 [(x-μ) ∂_σ ^2 q_θ]_a^b 
       + 2 (∂_σμ_B )^2  - 2 (μ - μ_B) ∂_σ ^2 μ_B.




  
First and second order partial derivatives of q_θ with respect to (μ,σ)
We also need to compute the partial derivatives of q_θ. For the first order, we get

    ∂_μ q_θ(x)    = q_θ(x) ( x-μ/σ^2 + [q_θ(x) ]_a ^b  ),
    ∂_σ q_θ(x)    = q_θ(x) (  1/σ[ (x-μ) q_θ(x) ]_a ^b -1/σ + (x-μ)^2/σ^3).

The computation of the second order derivatives follows

    ∂_μ ^2 q_θ(x)    = ∂_μ q_θ(x) ( x-μ/σ^2 + [q_θ(x) ]_a ^b  ) + q_θ(x) ( -1/σ^2 + [∂_μ q_θ(x) ]_a ^b  ),
    ∂_σμ ^2 q_θ(x)    = ∂_σ q_θ(x) ( x-μ/σ^2 + [q_θ(x) ]_a ^b  ) + q_θ(x) ( -2(x-μ)/σ^3 + [∂_σ q_θ(x) ]_a ^b  ),
    ∂_σ ^2 q_θ(x)    =  ∂_σ q_θ(x) ( 1/σ[ (x-μ) q_θ(x) ]_a ^b -1/σ + (x-μ)^2/σ^3) 
       + q_θ(x) ( -1/σ[ (x-μ) q_θ(x) ]_a ^b 
    +1/σ[ (x-μ) ∂_σ q_θ(x) ]_a ^b +1/σ^2 + -3(x-μ)^2/σ^4).
 

Let us summarize the previous computations:

    
  * we computed the FIM of the truncated normal family on [a,b];
    
  * we expressed the Christoffel symbols as a function of the conditional mean μ_[a,b] and conditional variance σ_[a,b]^2 (that are functions of θ) as well as their partial derivatives;
    
  * we then gave the explicit expression of these partial derivatives as a function of the truncated density q_θ on [a,b] and the partial derivatives of the latter;
    
  * lastly, we explicitly computed the partial derivatives of q_θ.

This concludes the computation of the FIM and Christoffel symbols of the truncated normal family on [a,b].





 §.§ Gumbel family

The Gumbel family is a location-scale family where the initial density p is given by p(x) = exp(-x - e^-x).


  §.§.§ Non-truncated case

The FIM for the non-truncated version is given by

    I_θ = 1/s^2[     1  γ -1;  γ -1 β + 1 ],

where γ≈ 0.5772... is the Euler-Mascheroni constant and β is given by the integral

    β = ∫_0^∞log(x)^2 x e^-xdx.

In this case, the Christoffel symbols can be easily computed. Indeed, they only depend on s. For the non-truncated Gumbel family, we numerically compute β using the Gaussian quadrature method of SciPy[For documentation, see https://docs.scipy.org/doc/scipy/tutorial/integrate.htmlhttps://docs.scipy.org/doc/scipy/tutorial/integrate.html].


  §.§.§ Truncated case

The density of the truncated Gumbel distribution on [a,b] writes q_θ(x) = 1/N_θ p_θ(x) 1_x∈ [a,b], where N_θ is the normalizing constant

    N_θ = ∫_a^b 1/s^2exp(-x-m/s - e^-x-m/s) dx = F(b-m/s) - F(a-m/s),

and F the cumulative distribution function of p given by
F(x) = exp(-e^-x).
In this case, we computed the FIM for the truncated family using the formula

    (I_θ^[a,b])_ij = - 𝔼_X ∼ q_θ[ ∂_θ_iθ_jlog q_θ(X) ],

where (θ_1,θ_2) = (m,s). For X ∼ q_θ, we have

    log q_θ(X) = -2log s - X-m/s - exp( -X-m/s) - log N_θ.

Taking the Hessian w.r.t. θ=(m,s) and integrating against q_θ leads to

    (I_θ^[a,b])_ij = -∂_θ_iθ_j   2log s - ∫_a^b ∂_θ_iθ_j[x-m/s] q_θ(x) dx 
       - ∫_a^b ∂_θ_iθ_j[exp( -x-m/s)] q_θ(x) dx - (Hess_θlog N_θ)_ij.
 
Notice that the first and forth term in the last equality can be explicitly computed. For the second and third ones, we need to resp. compute integrals of the form (after a variable substitution)

    ∫_c^d y exp( -y -e^-y)dy     and    ∫_c^d y^2 exp( -2y -e^-y)dy.

Here, c and d depend on a,b,m,s. We computed these integrals using the Gaussian quadrature method of SciPy. 
For the Christoffel symbols, we used the standard finite difference method for differentiating the coefficients of I_θ^[a,b] w.r.t. θ:

    ∂_m I_θ^[a,b]≈I_θ + (h,0)^[a,b] - I_θ^[a,b]/h   and   ∂_s I_θ^[a,b]≈I_θ + (0,h)^[a,b] - I_θ^[a,b]/h,

where we took h = 10^-7. Of course, more advanced numerical differentiation methods can be used instead.



 §.§ Gamma family

The pdf p_θ of the Gamma family writes

    p_θ(x) = β^α/Γ(α)x^α-1e^-β x,    θ = (α,β) ∈ (0,∞)^2.

Here, Γ is the Euler integral of the second kind. In this framework we do not consider a truncated version of this family. The FIM I_θ for this family can be easily computed. Indeed, we have

    I_θ = [ ψ'(α)  -1/β;  -1/β α/β^2 ].

Here, the function ψ is the digamma function. To numerically compute its first and second order derivatives, we used the previously discussed finite difference scheme with h=10^-7. 



§ A FEW ADDITIONAL ILLUSTRATIVE FIGURES FOR THE DENSITY PERTURBATION METHOD












plain





 Paper Title: On the mean-field limit of the Cucker-Smale model with Random Batch Method.




    
        
        
     
        a]Yuelin Wang sjtu_wyl@sjtu.edu.cn
        a]Yiwen Lin linyiwen@sjtu.edu.cn
[a]School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, P.R.China 


On the mean-field limit of the Cucker-Smale model with Random Batch Method
    [
    
==========================================================================



§ ABSTRACT 
    In this work, we focus on the mean-field limit of the Random Batch Method (RBM) for the Cucker-Smale model. Different from the classical mean-field limit analysis, the chaos in this model is imposed at discrete time and is propagated to discrete time flux. We approach separately the limits of the number of particles N→∞ and the discrete time interval τ→ 0 with respect to the RBM, by using the flocking property of the Cucker-Smale model and the observation in combinatorics. The Wasserstein distance is used to quantify the difference between the approximation limit and the original mean-field limit. Also, we combine the RBM with generalized Polynomial Chaos (gPC) expansion and proposed the RBM-gPC method to approximate stochastic mean-field equations, which conserves positivity and momentum of the mean-field limit with random inputs.

    
    Keywords: Random Batch Method, mean-field limit, Cucker-Samle model, stochastic Galerkin.






§ INTRODUCTION


    Collective behaviors of many-body systems are ubiquitous in the real world, like  flocking of birds <cit.>, swarming of fishes <cit.>, synchronicity of fireflies <cit.> and pacemaker cells <cit.>. We use the jargon “flocking" to describe the phenomenon in which self-propelled particles organize into an ordered motion using only limited environmental information and simple rules <cit.>. Since there is a great deal of literature on collective behaviors and related models, we refer the readers to <cit.> and the references therein.
    
     
    The Cucker-Smale model, proposed by Cucker and Smale <cit.>, is a famous model of collective behaviors that models phenomenologically  the flocking phenomenon. It is in the form of an N-body second-order system of ordinary differential equations or particle systems  for position and velocity, which resemble Newton's equations.  
    

     In addition, the mean-field limit, as the number of particles N →∞,  is also a prevalent tool for simplifying large particle systems, which makes use of continuum models derived within the framework of mathematical kinetic theory, to approximate collective behaviors. In these years, there has been extensive research on the mean-field limit of particle systems and we refer to <cit.> for survey articles.
     
    As a classical type of N-body systems, in the Cucker-Smale model, each particle interacts with N-1 particles, and thus the computational cost to solve it is 𝒪(N^2) per time step. An efficient algorithm to reduce computational complexity to 𝒪(N) is the Random Batch Method (RBM) proposed by Jin et al. <cit.> in 2020. The RBM indeed constructs a randomly decoupled system with subsystems of interaction between p particles, where the constant p≪ N. In this system, at each time step, the interaction only occurs within small batches with p particles. The choice of batches is random and thus the time-average effect makes it  a good approximation of the original system <cit.>.
    
     
    
    
    An interesting problem is to understand the mean-field limit of the RBM. In 2022, Jin and Li considered the mean-field limit of the RBM on first-order systems with Gaussian noise in <cit.>. 
        
        
        
        
    Inspired by <cit.>, in this paper, we investigate the mean-field limit of the RBM on the Cucker-Smale model as the number of particles tends to infinity. Unlike the first-order system considered in <cit.>, the Cucker-Smale model is a second-order system with interaction coupled with both velocity and position. Technically, moment control needs to be  derived by the asymptotic flock estimate instead of using the contraction property in first-order systems in <cit.>. Specifically, we mainly analyze the path from (b) to (c), and briefly discuss the path from (a) to (d) in the following figure (<ref>). On the upper left corner, Ha et al. give an estimate on (a) in <cit.> and a proof of the existence of mean-field limit on (d) in <cit.>. Their analysis on (d) is uniform in time but implicit in N. In later work <cit.>, an explicit estimate on N is given but only works on finite time. The four models in (<ref>) are represented in Section <ref>.
    
    
    Cucker-Smale system (<ref>) @ > (d)  N→∞ >> mean-field limit (<ref>)
    
             @A (a) τ→ 0 AA  @AA (c) τ→ 0 A
    Random Batch Cucker-Smale model (<ref>) @>> (b)  N→∞ > mean-field limit of RBM (<ref>)


    
   With the justification of (b) and (c),   we employ the RBM, which is an efficient particle-based numerical scheme for the approximation of stochastic mean-field equations of the Cucker-Smale model, inspired by <cit.>. We consider such an application in the stochastic setting, where the interacting force depends on a random variable modeling uncertainties.  This is an important problem in Uncertainty Quantification <cit.>. The method combines the RBM with a generalized Polynomial Chaos (gPC) expansion in the random space and thus we call it the RBM-gPC. It avoids loss of positivity like MCgPC in <cit.>, and moreover preserves the mean velocity during evolution.

    
    The rest of the paper is organized as follows. In Section <ref>, we give a concise introduction to the Cucker-Smale model including notations, assumptions and prerequisites. Section <ref> presents our main results on the analysis of limits (b) and (c) in (<ref>) with helpful discussions on the limits from (a) to (d). Section <ref> and Section <ref> provide the proof details. In Section <ref>, we show some numerical experiments with the RBM-gPC. Finally, Section <ref> is devoted to a summary of our main results and some remaining issues to be explored in the future.
    
    
    
    


  




§ PRELIMINARIES

In this section, we introduce the main models in (<ref>) with assumptions and notations. Also, some basic properties of the models are given.



 §.§ The Cucker-Smale model

Let X_i and V_i be the position and velocity of the i-th particle with unit mass, and ψ(| X_j -X_i|) be the communication weight between the j-th and i-th particles. The Cucker-Smale model reads as the following

    {d/dt X_i(t) =    V_i(t),   i=1,⋯, N,
    d/dt V_i(t) =   κ/N-1∑_j=1^N 
            ψ(|X_j(t)-X_i(t)|)(V_j(t)-V_i(t)),
        .

where κ is the nonnegative coupling strength and ψ satisfies  positivity, boundedness, Lipschitz continuity and mononticity conditions, i.e., there exist positive constants ψ_0, ψ_M  > 0  such that 

    0< ψ_0 ≤ψ(r) ≤ψ_M, ∀ r≥0;   ψ_Lip<∞;   (ψ(r_1)-ψ(r_2))(r_1-r_2)≤ 0,  r_1,r_2 ∈ℝ_+.

Without loss of generality, we set ψ_M = 1 in this paper for convenience. 
Under the condition (<ref>) of ψ, it is observed in <cit.> that the total momentum is conserved as a constant and the total energy is nonincreasing along the Cucker-Smale flow. Actually,

    supposing that {(X_i,V_i)} is the solution of  system (<ref>), then for any t>0, one has
    
    d/dt∑_i=1^N V_i (t) = 0,   d/dt∑_i=1^N |V_i(t)|^2 = -κ/N-1∑_i,jψ(|X_j(t) - X_i(t)|) |V_j (t) - V_i (t) |^2.




A key observation of the Cucker-Smale model, given by Ha et al. in <cit.>, is that the standard deviations of particle phase-space positions are dominated by SDDI (the system of dissipative differential inequalities):

    | d𝒳/dt| ≤𝒱,   d𝒱/dt≤ -ϕ (𝒳) 𝒱,

where (𝒳,𝒱) are nonnegative functions and ϕ is a nonnegtive measurable function. This is a useful tool to analyze the Cucker-Smale model. In this paper, we use a variant  of (<ref>) (Lemma <ref>) in the proof of our main result. One can find its proof in Lemma 3.1 in <cit.>.

    Suppose that two nonnegative Lipschitz functions 𝒳 and 𝒱 satisfy the coupled differential inequalities:
    
    | d𝒳/dt| ≤𝒱,   d𝒱/dt≤ -α𝒱 + γ e^-α t𝒳,    a.e.  t>0,

    where α and γ are positive constants. Then, 𝒳 and 𝒱 satisfy the uniform bound and decay estimates:
    
    
    𝒳(t) ≤2M/α (𝒳(0) + 𝒱(0)),   𝒱(t) ≤ M (𝒳(0) + 𝒱(0)) e^-α t/2,    t≥ 0,

    where M is given by
    
    M:= max{1,2γ/α e} + 8γ/α^3 e^3.





 §.§ The mean-field limit of the Cucker-Smale model


The corresponding evolution for the distribution function f(x, v, t) was first derived by Tadmor, etc. in <cit.> by the BBGKY hierarchy (e.g. <cit.>), which follows

    ∂_t f + v·∇_x f+ ∇_v · [ξ[f]f]=0,

where the operator ξ is defined by

    ξ[f](x,v,t) = κ∫∫ψ(|x-y|)(w-v)f(y,w,t)  dwdy.

It is clear that 

    ∂/∂ t∫∫ vfdxdv =0.

Similar with energy-decreasing in (<ref>) in the microscopic scale, the mean-field limit has the following property: 
 
Let f_0 be compactly supported in ℝ^2d with x = ∫ x f_0 (x,v) dxdv and v = ∫ v f_0 (x,v) dxdv. Define C_x0:=inf{X≥0  |  supp(f_0)⊂ B(x,X) ×ℝ^d } and C_v:=inf{V≥0  |  supp(f_0)⊂ℝ^d × B(v,V)}, where B(x,X) designates the ball of center x and radius X in ℝ^d. Let f_t be the unique solution of  (<ref>) with f(0) = f_0. If the initial data satisfies

    C_v < κ∫_C_x0^∞ψ (2s) ds,

then there exists C_x>0 such that 
    
    supp[f_t]⊂ B(x + tv, C_x) × B(v, C_v e^-κψ(2C_x) t).



    Under  assumption (<ref>) of ψ,  one has ψ(2C_x)≥ψ_0. Actually, the assumption of the lower bound ψ_0 is for the convenience of the proof and can be relaxed if (<ref>) is satisfied by both discrete and continuous systems, guaranteeing the flocking in the Cucker-Smale model.




 §.§ The Wasserstein distance


To introduce the topology of the space of probability measures, here we recall the definitions of the Wasserstein space <cit.>. For a metric space 
 (ℝ^d,ρ) and q≥1, define the Wasserstein space of order q by

    𝒫_q,ρ (ℝ^d):= {μ∈𝒫(ℝ^𝕕); ∫_ℝ^dρ(x,0)^q ν (dx)< ∞},

endowed with the Wasserstein metric 

    W_q,ρ(μ,ν) := ( inf_π∈Π (μ,ν){∫_ℝ^d ×ℝ^dρ(x,y)^q π (dx,dy)})^1/q,

where μ,ν∈𝒫_q (ℝ^d) and Π (μ,ν) is the set of probability measures on ℝ^d ×ℝ^d with marginals μ and ν respectively. The W_q metric, or distance, induces a kind of weak topology and measures the closeness between distributions. We use the notation W_q,p when ρ(x,y):= x-y _p and typically abbreviate W_q = W_q,2. 
For convenience in notation, we simplify · as |·| without ambiguity.


Recall the Jordan decomposition for a signed measure |μ| := μ^+ + μ^-. One can control the W_q distance by the weighted total variation by the following lemma (see e.g. Theorem 6.15 in <cit.>), which helps the proof of our main theorems, i.e., Theorem <ref> and Theorem <ref>, in Section <ref>.

    Let μ and ν be two probability measures on a Polish space (𝔛, d). Let q∈ [1,∞) and x_0 ∈𝔛. Then 
    
    W_q (μ,ν) ≤ 2^1-1/q( ∫ d(x_0,x)^q d |μ-ν|(x) )^1/q.





 §.§ The Random Batch method and its mean-field dynamics


In each sub-time interval [t_k-1,t_k), we set [i]_k as the batch containing the particle i. The RBM-(approximated) system of (<ref>) becomes:

    {∂_t X_i^R(t) =    V_i^R(t),   i=1,⋯, N,
    ∂_t V_i^R(t) =   κ/p-1∑_j∈[i]_kψ(|X_j^R(t)-X_i^R(t)|)(V_j^R(t)-V_i^R(t)),
            
        .

with initial data (X_i^R(0),V_i^R(0)) = (X_i^in,V_i^in), i = 1,⋯,N. See Algorithm <ref>.









In the N→∞ limit, the N-particle system is then reduced to a p-particle
subsystem described by the following system for t∈ [t_k,t_k+1):

    {∂_t X_i(t) =    V_i(t),   i=1,⋯, N,
    ∂_t V_i(t) =   κ/p-1∑_j=1^p 
            ψ(|X_j(t)-X_i(t)|)(V_j(t)-V_i(t)),
        .

with Z_i:=(X_i,V_i) and Z_i (t_k) being i.i.d drawn from f(·; t_k). Here f(·; t_k) is the law of Z_1(t_k^-) for k≥ 1 and f(·;0) equals the initial distribution f_0 . We impose Z_1 (t_k^-) = Z_1 (t_k^+).  Except the individual particle 1, the rest p-1 particles are sampled from an infinite pool of independent particles from particle 1 at each time step t_k. This process becomes the mean-field limit model of the RBM system (<ref>), and one may write the following mean-field limit for the RBM in terms of the probability distribution as shown in Algorithm <ref>, while (<ref>) becomes its microscopic description.










§ THE MAIN RESULT


In this paper, we show the conservation law and the mean-field limit of the Random-Batch Cucker-Smale model. First, we show the conservation of momentum and dissipation of kinetic energy of the  mean-field limit system of RBM in Section <ref>. Then, we analyze the procedure (b) and (c) in Section <ref>. 


Assume that the initial data of Algorithm <ref> and Algorithm <ref> obey the same law f_0(x,v)dxdv. For notational convenience, recall x:= ∫ x f_0(x,v) and v:=∫ vf_0 (x,v); and set z_i:=(x_i,v_i) for each i and 

    f(x,v;t):= ∫_ℝ^2d(p-1) f^(p)(⋯ ; t^-)dx_2 ⋯ dx_p dv_2⋯ dv_p = f_t(x,v).

Note that f is defined on ℝ^2d in Algorithm <ref> and f^(p) is defined on ℝ^2dp. Also, the definition of f is consistent with that in Algorithm <ref> at t=t_k.

In the following, we suppose that f_0 is compactly supported and then there exists a constant R>0 such that supp[f_0]⊂ B(0,R), where B(0,R) is a ball with radius R centered in the origin. Without loss of generality, we set v=0 as a standardization of the initial momentum.




 §.§ Basic characteristics of the mean-field limit system of RBM


Here, we show that the two basic characteristics: conservation of momentum and dissipation of kinetic energy, as observed in the Cucker-Smale model (<ref>) and the RBM system (<ref>), also holds for the RBM mean-field system described in Algorithm <ref>.

 
For t∈ (t_k,t_k+1), it holds
    
    ∂/∂ t∫_ℝ^2d v f(x,v;t) dx dv = 0.

In particular, since v=0, ∫ vf(x,v;t) = 0 for any t∈ [0,T].



By the definition of f, one has

    ∂/∂ t∫_ℝ^2d vf(x,v;t)dxdv =    ∫_ℝ^2d∫_ℝ^2d(p-1) v ∂/∂ t f^(p) (x,·, v, ·;t^-)dxdx_2⋯ dx_p dvdv_2⋯ dv_p
    
            =    ∫_ℝ^2dp v_1 [ -∑_i=1^p ∇_x_i· v_i f^(p)(⋯ ; t) - ∑_i=1^p ∇_v_i·ξ_i f^(p)(⋯ ;t)]dz_1⋯ dz_p
    
            =    κ/p-1∑_j=2^p∫_ℝ^2dpψ (|x_j - x_1|)(v_j-v_1) f^(p)_t(z_1,⋯ ,z_p)dz_1⋯ dz_p
    
            =     0.

The last equality is derived from the fact that ψ (|x_j - x_1|)(v_j-v_1)f^(p)_t is an odd function on (z_j,z_1).





Note that f(x,v;t_k^-) = f(x,v;t_k) = f(x,v;t_k^+). Then by induction one has ∫ vf(x,v;t) = v.


 
For any t∈ (t_k,t_k+1),
    
    ∂/∂ t∫_ℝ^2d |v|^2 f(x,v;t)dxdv ≤ 0.

In particular, M_2^v f_t ≤ M_2^vf_0, where M_2^v f_t:=∫_ℝ^2d |v|^2 f(x,v;t)dxdv. Moreover, since 0<ψ_0≤ψ, it holds
    
    M_2^v f_k+1≤ M_2^v f_0(e^-2κψ_0 τ + 8κ^2 ψ_0/p-1τ^2 )^k .



    
    For any t∈ (t_k,t_k+1),
        
    ∂/∂ t M_2^v f_t =    ∫_ℝ^2dp |v_1|^2 ( -∑_i=1^p ∇_x_i· (v_i f^(p)_t) - ∑_i=1^p ∇_v_i· (ξ_i f^(p)_t) )
    
                    =   ∫_ℝ^2dpκ/p-1∑_j=1^pψ(|x_1-x_j|)(v_j-v_1)· 2v_1 f^(p)_t
    
                    =    -∫_ℝ^2dpκ/p-1∑_j=1^pψ(|x_1-x_j|)|v_j-v_1|^2 f^(p)_t
    ≤    0.

        The last equality is derived by the symmetry of ψ and f_t^(p). By the continuity of f, one has M_2^v f_t ≤ M_2^vf_0. Since ψ has a positive lower bound ψ_0, furthermore, it holds
        
    ∂/∂ t M_2^v f_t ≤    -∫_ℝ^2dpκ/p-1∑ _j=1^p ψ_0 |v_j-v_1|^2 f^(p)_t
    
                    =    -2κψ_0 ∫_ℝ^2d |v|^2 f_t(x,v) dxdv + 2κψ_0 ∫_ℝ^2dp v_1· v_2 f^(p)_t,

        by the symmetry of f^(p)_t. For the second term in (<ref>), one has
        
    ∂/∂ t   ∫_ℝ^2dp v_1 · v_2 f^(p)_t
    
                    =    κ∫_ℝ^2dpv_2/p-1·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t + v_1/p-1·∑_k 2^p ψ(|x_2-x_k|)(v_k-v_2) f^(p)_t
    
                    =    2κ∫_ℝ^2dpv_2/p-1·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t.

        For the last integration in (<ref>), by the indistinguishability of the particles, one has  
        
    ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t
       = ∫_ℝ^2dp v_2·∑_k 1,2^p ψ(|x_1-x_k|)(v_1-v_k)f^(p)_t + ∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       = ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_1-v_k)f^(p)_t 
                    +∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       + ∫_ℝ^2dp v_2·ψ(|x_1-x_2|)(v_2-v_1)f^(p)_t
       = -∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t +∫_ℝ^2dp |v_2-v_1|^2 ψ(|x_1-x_2|) f^(p)_t.

        Hence, recalling ψ≤ 1, it holds
        
    ∫_ℝ^2dp v_2·∑_k 1^p ψ(|x_1-x_k|)(v_k-v_1)f^(p)_t
            = 1/2∫_ℝ^2dp |v_2-v_1|^2 ψ(|x_1-x_2|)f^(p)_t≤ 2 M_2^v f_t.

        Then it yields from (<ref>) that 
        
    ∂/∂ t∫_ℝ^2dp v_1 · v_2 f^(p)_t≤4κ/p-1 M_2^vf_t_k-1.

        By combining (<ref>) and (<ref>), it holds
        
    ∂/∂ tM_2^vf_t    ≤ -2κψ_0 M_2^vf_t + 2κψ_04κτ/p-1M_2^vf_t_k-1 +  2κψ_0∫ v_1 · v_2 f_t_k-1^(p)
       =  -2κψ_0 M_2^vf_t + 2κψ_04κτ/p-1M_2^vf_t_k-1
       ≤  -2κψ_0M_2^v f_t + 8κ^2ψ_0/p-1τ M_2^vf_t_k-1.

        By Grönwall's inequality, one has
        
    M_2^v f_k≤ M_2^v f_t_k-1(e^-2κψ_0 τ + 8κ^2ψ_0/p-1τ^2 ) ≤ M_2^v f_0(e^-2κψ_0 τ + 8κ^2 ψ_0/p-1τ^2 )^k .

    
    


    
    
    
    
    
    
    
    
    
    
    
    
    





 §.§ The main theorems



First we discuss the Wasserstein distance between the distribution of the RBM of the N-particle system (<ref>) and the distribution of the N→∞ system (<ref>). 

Define the operator 𝒬_N^(k) on the probability measure space 𝒫(ℝ^2d) as follows. Let Z_i^R(0) be i.i.d drawn from f_0. Corresponding to Algorithm <ref>, we define 

    𝒬_N^(k)(f_0):= Law(Z_1^R(t_k)),

where Law(Z_1^R) means the law of Z_1^R.  Conditioning on a specific sequence of random batches, the particles are not exchangeable. However, when considering the mixture of all possible sequences of random batches, the laws of Z_i^R(t_k) are identical.

In addition, we define 𝒬_∞ as one time step evolution in Algorithm <ref>. Thus one arrives at 

    𝒬_∞^k (f_0) = 𝒬_∞∘⋯∘𝒬_∞(f_0),  (k copies),

which is expected to be the mean field limit of the RBM after k steps. Note that the 𝒬_∞ dynamics can fully determine the probability transition, knowing the marginal distribution of Z_1, while knowing only the marginal distribution is not enough for the dynamics of 𝒬_N^(k). The joint distribution must be known in the latter case. 


Theorem <ref> gives an estimate between 𝒬_∞^k (f_0) and 𝒬_N^(k)(f_0) on the W_q distance. In this theorem, we introduce a quantity ϵ_k∈ℝ, which was first introduced in <cit.>. Plainly speaking, the ϵ_k means the probability that the particle i is not “clean", where a particle i is clean at t_k^- means its batchmates at t≤ t_k were mutually independent and independent to the particle i when they interacted. For better organization, we give the definition of "clean"   and more details in Section <ref>.


    For f_0 with the compact support supp [f_0], it holds that 
    
    W_q(𝒬_∞^k (f_0),𝒬_N^(k) (f_0))≤ C (1+t_k) ϵ_k^1/q,

    with q ∈ [1,∞) and C=C(κ,ψ_0,R). For fixed k, it holds
    
    lim_N→∞ϵ_k =0,

    with ϵ_k≤𝒪(N^-1).







    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    



    Then, we consider the limit dynamics given by the operator 𝒬_∞ and how 𝒬_∞^n (f_0) approximates the dynamics of the Fokker-Planck equation (<ref>), i.e. the procedure (c) in (<ref>). We prove that the W_q distance can be controlled by τ with a smooth initial distribution f_0 with a compact support uniformly in time. 
    
     
        Let f be a solution to the Fokker-Planck  equation (<ref>) with initial condition f(0) = f_0. 
        It holds
        
        
        
        
        the uniform-in-time estimate
        
    sup_n W_q (𝒬_∞^n (f_0), f(nτ))≤ Cτ,

        with q∈[1,∞)and C=C(κ,ψ_0,R).
    

    







 §.§ Some helpful discussions



In addition, we give some  discussions on Lemma <ref> that will be used for the proof of Theorem <ref>, and on the commutation of (<ref>).





  §.§.§ Approximation using PDE analysis


Similar with [<cit.>, Lemma 4.4], one can obtain an 𝒪(τ^2/q) bound on the W_q distance between 𝒬_∞ (f_t_k) and S(f_t_k) in Proposition <ref>. Here S(f_t_k) denotes the law of (<ref>) in Section <ref>.


     Let f be a solution to the Fokker-Planck  equation (<ref>) with initial condition f(0) = f_0 ∈ (C^2∩ W^2,∞)(ℝ^2d). Suppose that the interaction kernel ψ satisfies 
    sup_r≥ 0 (|ψ^'(r)| + |ψ^''(r)|) ≤ϵ_ψ
 with a constant ϵ_ψ. It holds
    
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ,ϵ_ψ , R, d) τ^2/q,    q∈ [1,∞).


The proof relies on the PDE analysis on the evolution equations on 𝒬_∞ (f_t_k) and S(f_t_k), using the corresponding operator groups. There are two limitations on this method. One is that more regular assumptions on both the interaction kernel and initial data are needed to guarantee the existence of a unique classical solution f_t ∈ C^2([0,T)×ℝ^2d) for (<ref>). In addition, if one uses the estimate obtained by Lemma <ref> with the total variance bound, the order of τ is 2/q. 
A detailed proof of Proposition <ref> is given in Appendix <ref>. 





  §.§.§ The mean-field limit of the Cucker-Smale system
 

Up to now, we have given a brief analysis of the path ((b),(c)) in the lower right corner of (<ref>). Here, we discuss the upper left corner path in the sense of the Wasserstein distance as well. Denote by 

    μ_t^N:=∑_i=1^N 1/Nδ_z_i(t)
 the empirical measure and μ_t^R the empirical measure derived by the RBM with μ_0^R = μ_0^N.



 
    For the path from (a) to (d) in (<ref>), it holds 
    
    𝔼 W_2 (f_t,μ_t^R)     ≤ C(κ, R,ψ)√(τ/p-1) +
            C(R)·{   N^-1/4     if  d<4,
       N^-1/4√(log(1+N))     if  d=4,
       N^-1/d     if  d>4.
            .



    We first consider the procedure (a). For the first τ→ 0 limit, by the definition of the W_q distance, one has
    
    W_2^2 (μ_t^R, μ_t^N) 
            ≤1/N∑_i=1^N (|X_i^R - X_i|^2 + |V_i^R - V_i|^2).

    Based on [<cit.>, Theorem 3.2], one has
    
    𝔼(1/N∑_i-1^N |V_i^R (t) - V_i (t)|^2)
            ≤ C(1/p-1-1/N-1)τ + Cτ^2 + C(1+τ) exp(-κψ_0 t),

    for some constant C depending on κ, R and ψ.
    Note that
    
    d/dt∑_i=1^N |X_i^R - X_i|^2 ≤
        2√(∑_i=1^N |X_i^R - X_i|^2)√(∑_i=1^N |V_i^R - V_i|^2),

    by the Cauchy-Schwarz inequality and
    
    √(∑_i=1^N |X_i^R - X_i|^2)≤√(∑_i=1^N |V_i^R - V_i|^2) T.

    It thus holds
    
    𝔼 W_2 (μ_t^N,μ_t^R) ≤𝔼1/√(N)√(∑_i=1^N |X_i^R - X_i|^2 + ∑_i=1^N |V_i^R - V_i|^2)
    ≤√(C τ(1/p-1-1/N-1))(1+T).
.

    Next, for the procedure (d), by the estimate of [<cit.>, Corollary 1], one has W_q (μ_t^N,f_t) ≤ C(κ,R)W_q(μ_0^N,f_0). 
    In addition, according to the convergence rate of the empirical measure in [<cit.>, Theorem 1], with f_0 compactly supported, one has 
    
    𝔼 W_q^q (f_0,μ_0^N) ≤ C(R)·{   N^-1/2     if  q>d/2,
       N^-1/2log(1+N)      if  q=d/2,
       N^-q/d     if  q<d/2.
        .


    Therefore, under the same assumption of Theorem <ref> and Theorem <ref>, for the path from (a) to (d), one has 
    
    𝔼 W_2 (f_t,μ_t^R)    ≤𝔼[W_2 (μ_t^N,μ_t^R) + W_2 (f_t,μ_t^N)]    
             ≤ C(κ,R,ψ)√(τ/p-1) + C(κ,R) 𝔼 W_2 (f_0,μ_0^N)
       ≤ C(κ,R,ψ)√(τ/p-1) + C(κ,R)(𝔼 W_2^2 (f_0,μ_0^N))^1/2.











§ PROOF OF THEOREM <REF>


    In this section, we prove Theorem <ref> by an observation in combinatorics. 
    Recalling the critical quantity ϵ_k we mentioned above Theorem <ref>, we first give the Lemmas <ref>-<ref>, for the proof of Theorem <ref>, and show the mathematical definition of ϵ_k in Section <ref>.
    The proof of Lemmas <ref>-<ref> are shown in Sections <ref>-<ref> respectively.
    
    Finally, Theorem <ref> will be proved in Section <ref> by combining Lemma <ref>, Lemma <ref> and an estimate on the diameters of the compact support in space and velocity, shown in Lemma <ref>.


    
    
        Under the assumption and notation of Theorem <ref>, it holds
            
    |𝒬_∞^k (f_0) - 𝒬_N^(k) (f_0)|_TV≤ 2ϵ_k,

        where ϵ_k is a positive constant  with respect to N.
    
    
    
    
        For any fixed k, with the assumption and notation of Theorem <ref>, it holds that ϵ_k ≤𝒪(N^-1).
    


    




 §.§ Prerequisites

    To give a detailed explanation of ϵ_k in Theorem <ref>, we introduce the concept “clean” in Definition <ref>. First, for each particle i, we define a sequence of lists {L_i^(k)} associated with i, given as follows:
    
        
  * L_i^(0) = {i};
        
  * For k≥ 1, let 𝒞_q^(k-1) be the batch that particle i stays in for t∈[t_k-1,t_k), then
        
    L_i^(k) = ∪_j∈𝒞_q^(k-1) L_j^(k-1).

    
    Here, L_i^(k) can be viewed as the particles that have impacted i for t < t_k. Clearly, a particle i_1∈ L_i^(k) might not have been a batchmate of i. It could have been a batchmate of i_2, and then i_2 was a batchmate of i at some time. The important observation is that if L_i^(k) and L_j^(k) do not intersect for a given sequence of random batches, then particles i and j are independent at t_k^-. Note that we are not claiming all particles in L_j^(k) are independent of those in L_j^(k) at t_k^-. In fact, it is possible that some i_1 ∈ L_i^(k) and j_1 ∈ L_j^(k) are in the same batch on [t_k-1, t_k). However, i_1 and j_1 must be independent at the times when they were added to the batches that eventually impact i,j at t_k^-. Then we are motivated to define the following.
    
    
        We say the particle i is clean on [t_k,t_k+1) if the batch 𝒞_q^(k) that contains i at t_k^+ satisfies the following:
        
            
  * any j∈𝒞_q^(k) is clean at t_k^-;
            
  * any j,ℓ∈𝒞_q^(k) with jℓ, L_j^(k) and L_ℓ^(k) do not intersect.
        
    
    In other words, a particle i is “clean" at t_k^- if its batchmates at t<t_k were mutually independent and independent to i when they interacted.
    
   
    Let A_k denote the set of particles that are clean at t_k^-. Then
    
    A_0 = {1,⋯, N}.

    For k≥ 1, one has 
    
    A_k = { i ∈ A_k-1: i∈𝒞_q^(k-1), ∀ j,ℓ∈𝒞_q^(k-1),  jℓ ,
    
            j∈ A_k-1, ℓ∈ A_k-1,  L_j^(k-1)∩ L_ℓ^(k-1) = ∅}.

    Now we define
    
    ϵ_k := ℙ(1∉ A_k).

    Note that by symmetry, ϵ_k is also the probability that particle i is not clean.









 §.§ Proof of Lemma <ref>


    We first introduce Lemma <ref>.
     
        Consider a fixed sequence of divisions of random batches {𝒞^(ℓ)}_ℓ≤ k-1.
        
            
  * It holds that
                
    |L_i^(k)|≤ p^k,

                and the particle i is clean at t_k^- if and only if the equality holds.
            
  * The distribution of Z_i (Recall Z_i := (X_i,V_i).) for a clean particle i at t_k^- is 𝒬_∞^k (f_0).
        
        Here the symbol |A| means the cardinality of a set A.
    
    
        The proof is a straightforward induction. Here, let us just briefly mention the proof of the second claim. For k=0, the statement is trivial. Now, suppose the statement is true for all k ≤ m-1. We now consider k = m.
        
        For the given sequence of random batches {𝒞^(ℓ)}_ℓ≤ k-1, that a particle i is clean at t_m^- means that on [t_m-1, t_m), the particles in the batch for i are independent of i at t_m-1. By the induction assumption, the distribution of one particle at t_m-1 is given by 𝒬_∞^m-1(f_0). By the independence, the joint distribution of them at t_m-1 is therefore
        
    f^(p) (⋯, t_m-1) = 𝒬_∞^m-1(f_0)^⊗ p.

        From t_m-1 to t_m, the evolution of the joint distribution obeys  equation (<ref>). Hence, at t_m^-, the distribution of particle i is given by 𝒬_∞^m(f_0). 
    



    Now we begin to prove Lemma <ref>.

    
        By Lemma <ref>, one has
        
    𝒬_N^(k) (f_0) = ℙ(1∈ A_k)𝒬_∞^k (f_0)
                + ℙ(1∉ A_k)ν_k,

        for some probability measure ν_k. To see this, consider all possible sequences of random batches. Only the first k divisions of batches (i.e. ones at t_0,⋯, t_k-1) will affect the distribution at t_k. This subsequence (the first k divisions) can take only finitely many values, and let {c^ℓ}_ℓ≤ k-1 be such values. Then, for any E⊂ℝ^d that is Borel measurable,
        
    𝒬_N^(k) (f_0)[E] = ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1}ℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)
                ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ).

        Lemma <ref> tells us if {c^ℓ}_ℓ≤ k-1 is a value such that particle 1 is clean, then
        
    ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ) = 𝒬_∞^k (f_0)[E].

        Hence,
        
    𝒬_N^(k) (f_0)[E]    =  ℙ(1∈ A_k)𝒬_∞^k (f_0)[E] 
         + ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1},  1∉ A_kℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)
                ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 )
       = ℙ(1∈ A_k)𝒬_∞^k (f_0)[E] + ℙ(1∉ A_k)ν_k (E),

        
        
        
        
        
        
        with
        
    ν_k (E) = ∑_{𝒞^ℓ = c^ℓ , ℓ≤ k-1},  1∉ A_kℙ(𝒞^ℓ = c^ℓ ,  ℓ≤ k-1)/ℙ(1∉ A_k)ℙ(Z_1 ∈ E  | 𝒞^ℓ = c^ℓ ,  ℓ≤ k-1 ).

        Clearly, ν_k is a convex combination of some conditional marginal distributions of Z_1, each being Law(Z_1) conditioning on a particular sequence of batches for {1∉ A_k}. Hence ν_k is a probability measure. 
        
        By (<ref>), it holds that
        
    |𝒬_∞^k(f_0) - 𝒬_N^(k) (f_0)| ≤ 
                (1-ℙ(1∈ A_k))𝒬_∞^k (f_0) + ℙ(1∉ A_k)ν_k
                = ϵ_k (𝒬_∞^k (f_0) + ν_k).

        Therefore, the total variation distance between the two measures is controlled by
        
    |𝒬_∞^k(f_0) - 𝒬_N^(k) (f_0)|_TV≤ 
                (1-ℙ(1∈ A_k)) + ℙ(1∉ A_k) = 2ϵ_k.

    




    
    
    




 §.§ Proof of Lemma <ref>

    Furthermore, we provide the proof of Lemma <ref> which is a simple variant of [<cit.>, Theorem 3.2].

    
        First of all, clearly, one has
        
    ϵ_0  = 1-1 = 0.

        Now, one can do induction on k. Assume
        
    lim_N→∞ϵ_k = 0,

        and then consider the batches for t_k → t_k+1^-. Assume the batch for particle 1 is 𝒞_q^(k). Denote
        
    B_k = {∀ j, ℓ∈𝒞_q^(k),  jℓ,   L_j^(k)∩ L_ℓ^(k) = ∅}.

        Let ℬ := 𝒞_q^(k)\{1} be the set of other particles that share the same batch with particle 1. Then, by definition of A_k+1, it holds
        
    ℙ (1∈ A_k+1) = ∑_j_1,⋯, j_p-1ℙ(ℬ={j_1,⋯, j_p-1})×
    ℙ(B_k ∩{1∈ A_k}∩_ℓ =1^p-1{j_ℓ∈ A_k} | ℬ= {j_1,⋯, j_p-1}).

         Denote E:= {ℬ={j_1,⋯, j_p-1}}, where we omit the dependence in j_ℓ, 1≤ℓ≤ p-1 for notational convenience. Conditioning on B_k ∩ E (i.e., provided that the event B_k ∩ E happens), whether the particles are clean or not is independent. Hence,
         
    ℙ(B_k ∩{1∈ A_k}∩_ℓ =1^p-1{j_ℓ∈ A_k} |  E) 
    
                 = ℙ(B_k | E))ℙ({1∈ A_k}∩_ℓ = 1^p-1{j_ℓ∈ A_k}|E,B_k)
    
                 = ℙ(B_k|E)Π_ℓ=1^pℙ(j_ℓ∈ A_k|E,B_k),

         where one has set j_p =1. Moreover,
         
    ℙ(j_ℓ∈ A_k|E,B_k) = ℙ({ j_ℓ∈ A_k }∩ E∩ B_k)/ℙ(E∩ B_k) = ℙ({1 ∈ A_k }∩ E∩ B_k)/ℙ(E∩ B_k) = ℙ({1 ∈ A_k }∩ B_k)/ℙ(B_k).

         The second and the last equalities are due to symmetry. For the last equality, ℙ({ℬ = {j_1⋯,j_p-1}}∩ B_k) should be equal for all possible j_1,⋯,j_p-1, and the same  is true for the numerator. This actually is a kind of independence. Hence, eventually due to the fact
         
    ∑_j_1,⋯, j_p-1ℙ (ℬ = {j_1,⋯, j_p-1}) ℙ(B_k | ℬ = {j_1,⋯, j_p-1}) = ℙ (B_k),

        one has
        
    1-ϵ_k+1 = ℙ (1∈ A_k+1) ≥ℙ (B_k) (1-ϵ_k 
                )^p.

        Hence, it suffices to show
        
    lim_N→∞ℙ (B_k) =1.

        To get an estimate for this, we consider the following equivalent way to construct L_i^(k): one starts with L_i ←{i} and repeat the following k times:
        
            
  * Set L_tmp← L_i and A=∅.
            
  * Loop the following until L_tmp is empty.
            
                
  (a) Pick a particle i_1 ∈ L_tmp, then choose p-1 particles from {1,⋯,N}\ A ∪{i_1} denoted by { i_2,⋯ , i_p}.
                
  (b) Set L_i← L_i ∪{ i_2, ⋯, i_p}.
                
  (c) Set A← A ∪{ i_1, i_2, ⋯, i_p}.
                
  (d) Set L_tmp← L_tmp\{ i_1, i_2, ⋯, i_p}.
            
        
        In the above, we are actually looking back from t_k-1. In the j-th iteration, we are constructing batches at t_k-j. Hence, this is an equivalent way to construct L_i^(k).

        Now, we estimate ℙ(B_k) by constructing the lists L_j_ℓ^(k): 1≤ℓ≤ p for j_ℓ∈𝒞_q^(k) using the above procedure. Consider that the lists for j_1,⋯,j_ℓ -1 have been constructed, which have included at most (ℓ -1)p^k particles. Now, for L_j_ℓ^(k) not to intersect with the previous lists, one has to choose particles from {1,⋯,N}\ [∪_z=1^ℓ -1 L_j_z^(k)∪ A ∪{i_1}] in 2.(a) step. Conditioning on the specific choices of L_j_ℓ^(k): 1≤ℓ≤ p and A with 
        
    N_1:= |L_j_ℓ^(k)∪ A∪{i_1}|,   N_2:= |A|,

        this probability is controlled from below by
        
    N- N_1p-1/N-1- N_2p-1≥N- 1-ℓ p^kp-1/N-1p-1.

        Hence, as N→∞,
        
    ℙ(B_k)≥Π_ℓ=1^p [N- 1-ℓ p^kp-1/N-1p-1]^k = 1- 𝒪(N^-1).

        Hence, lim_N→∞ϵ_k+1=0 and the claim follows.
    
    





 §.§ Proof of Theorem <ref>

    


    
    
    Now we can prove Theorem <ref>.
    
    Recall that the TV distance is defined by
    
    |μ - ν|_TV := sup_A∈ℱ |μ(A) - ν(A)|,

    for two probability measures μ, ν defined on certain sample space Ω with events ℱ. 


    We denote the diameters of the compact support in the spatial and velocity variables at time t by 𝒟_X(t) and 𝒟_V(t), respectively, i.e.
    
    𝒟_X(t) := max_i,j| X_i-X_j|,   𝒟_V(t) := max_i,j| V_i-V_j|.

    
    For the RBM system (<ref>), Ha et al. <cit.> gave an estimate on 𝒟_X^R and 𝒟_V^R shown in Lemma <ref>. This estimate works on any fixed sequence of batches and we provide the proof in Appendix <ref> for completeness.
    




        
        Let X^R,V^R be a solution of the RBM model (<ref>) with initial data f_0 satisfying 
        
    𝒟_X(0) + 𝒟_V(0)<∞.

        Then one has
        
    𝒟_V^R (t) ≤𝒟_V(0),   𝒟_X^R (t)≤𝒟_X(0) + 𝒟_V(0)t,   ∀ t ≥ 0.

    



    
    

    
        Denote z:= (x,v), Z^R_i:=(X_i^R,V_i^R), and |z|^q:=|x|^q + |v|^q.
        By Lemma <ref>, taking μ = 𝒬_∞^k (f_0) and ν = 𝒬_N^(k)(f_0), one has
        
    W_q (𝒬_∞^k (f_0),𝒬_N^(k)(f_0)) ≤ 2^1-1/q(∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z))^1/q.


       Note that Lemma <ref> degenerates into the original Cucker-Smale model when p=N. Hence for t∈[0,t_1), any {Z_i^(0)(t)}_i=1^p satisfying (<ref>), one obtains
       
    𝒟_V^(0)(t)≤𝒟_V(0),   𝒟_X^(0) (t) ≤𝒟_X (0) + 𝒟_V(0)t.

       Also, by the assumption of f_0, one has v=0, and therefore x=0. Since (<ref>) holds independent of the choice Z_i, defining 
       
    𝖷(t) :=   max{ |x|  |∃ v,  s.t.  f(x,v;t) >0},
    𝖵(t) :=   max{ |v|  |∃ x,  s.t.  f(x,v;t) >0},

       one has for t∈[0,t_1],
       
    𝖵(t)≤ C,   𝖷(t)≤ C(1+t),

       where C is a constant depending on R. 
       Then for t∈[t_1,t_2], for the new subsystem {Z_i^(1)}_i=1^p with initial distribution f_t_1, one can use Lemma <ref> again to get
       
    𝒟_V^(1)(t)   ≤𝒟_V^(1)(0)≤ 2𝖵(t_1) ≤ C, 
    𝒟_X^(1) (t)    ≤𝒟_X^(1) (0) + 𝒟_V^(1)(t-τ)
               ≤ 2𝖷(t_1) + 2𝖵(t_1)(t-τ)≤ C(1+t),

       and (<ref>) holds for t∈[t_1,t_2].
       By induction, for any t, (<ref>) holds. Therefore |𝖷(t_k)|^q + |𝖵(t_k)|^q ≤ C(1+t_k)^q.       
       
       Also, by Lemma <ref>, it holds |Z^R_i|^q≤ C(1+t_k)^q.
       Then for any z∈supp[| 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|], |z|^q≤ C(1+t_k)^q. Therefore, 
       
    ∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z) 
               
               
               
               ≤  C(1+t_k)^q | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|_TV.

       
       By Lemma <ref>, one has 
       
    ∫_ℝ^2d |z|^q d | 𝒬_∞^k (f_0)-𝒬_N^(k)(f_0)|(z) ≤ C(1+t_k)^q ϵ_k.

       Hence W_q (𝒬_∞^k (f_0),𝒬_N^(k)(f_0)) ≤ C(1+t_k)ϵ_k^1/q. Then by Lemma <ref>, we finish the proof.
    


    
        The above analysis is not enough to get the mean-field limit independent of τ and for fixed N, ϵ_k→ 1 as k→∞. This is a limitation of the technique in the paper.
    














§ PROOF OF THEOREM <REF>


    Here gives an outline of the proof of Theorem <ref>. First, we show the stability of 𝒬_∞ in Section <ref>. Second, an auxiliary system is introduced in Section <ref> as a bridge to estimate the W_q distance between 𝒬_∞(f_t_k) and f_t_k+1. 
    
    Last,  the proof of Theorem <ref> is given in Section <ref>.




 §.§ Stability of 


In Lemma <ref> and Lemma <ref>, we give the stability analysis on the operator 𝒬_∞.
 
    Suppose two probability measures μ_1,μ_2 satisfying ∫ vdμ_1=∫ vdμ_2, with compact supports supp[μ_1] , supp[μ_2] respectively, where supp[μ_i]⊂ℝ^d× B(0,R_μ), i=1,2, and the constant R_μ >0. Then it holds
    
    W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤ CW_q(μ_1,μ_2),    q=[1,∞),

    where C(κ,ψ,R_μ) is a positive constant.



    Set (X_i,V_i)(0)∼μ_1, and (X_i,V_i)(0)∼μ_2, for i=1, ⋯, p. The key to the proof is to make use of Lemma <ref> to get the contraction property of the velocity difference. 
    
    The two steps are as follows.
    

  
Step 1.
    First, we show an auxiliary estimate on 𝒟_V(t). By following the formulation of [<cit.>, Lemma 2.2], denote
    
    Ψ_ij:= ψ(|X_i-X_j|)/p + ( 1- ∑_j=1^p ψ(|X_i-X_j|)/p)δ_ij,

    where δ_ij is the Kronecker delta. Then Ψ_ij is nonnegative and satisfies
    
    Ψ_ij≥ψ(|X_i -X_j |)/p,   ∑_j=1^p Ψ_ij =1,  ∑_j=1^pΨ_ij(V_j-V_i) = ∑_j=1^pψ(|X_i -X_j |)/p(V_j-V_i).

    We denote V_i = (V_i^k) and obtain the following equality
    
    1/2d/dt   |V_i - V_j|^2
            = 1/2d/dt∑_k=1^d|V_i^k - V_j^k|^2
    
            =   ∑_k=1^d(V_i^k - V_j^k)(V_i^k - V_j^k)
    
            =   ∑_k=1^d κ p/p-1 (V_i^k - V_j^k)[
            ∑_m=1^p Ψ_im(V_m^k - V_i^k) - ∑_m=1^p Ψ_jm(V_m^k - V_j^k)]
    
            =    -∑_k=1^d κ p/p-1 (V_i^k - V_j^k)^2
            +  ∑_k=1^d κ (V_i^k - V_j^k)( ∑_m=1^p (Ψ_im-Ψ_jm)V_m^k)
    
            =    -∑_k=1^d κ p/p-1 (V_i^k - V_j^k)^2
       +  ∑_k=1^d κ p/p-1 (V_i^k - V_j^k)[ ∑_m=1^p (Ψ_im-min{Ψ_im,Ψ_jm} + min{Ψ_im,Ψ_jm} - Ψ_jm)V_m^k].

    For a given time t, one can choose indices i_t and j_t satisfying the relation 
    
    𝒟_V(t) = |V_i_t(t) -V_j_t(t)|,

    and thus for each m=1,2,⋯,p,
    
    ∑_k=1^d (V_i_t^k -V_j_t^k)V_j_t^k ≤∑_k=1^d (V_i_t^k -V_j_t^k)V_m^k ≤∑_k=1^d (V_i_t^k -V_j_t^k)V_i_t^k.

    Combining (<ref>) and (<ref>), one obtains
    
    1/2d/dt|V_i_t - V_j_t|^2       
            ≤     -∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)^2
       +  ∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)V_i_t^k ∑_m=1^p (Ψ_im-min{Ψ_im,Ψ_jm}) 
       -  ∑_k=1^d κ p/p-1 (V_i_t^k - V_j_t^k)V_j_t^k ∑_m=1^p (Ψ_jm -min{Ψ_im,Ψ_jm})
    
            =   -∑_m=1^pκ p/p-1min{Ψ_im,Ψ_jm}|V_i_t - V_j_t|^2
    ≤    - κ p/p-1ψ_0 |V_i_t - V_j_t|^2.

    Note that 𝒟_V is Lipschitz continuous with respect to t and thus it is differentiable almost everywhere. Consider the instant t at which 𝒟_V is differentiable. Then,
    
    d/dt𝒟_V^2 ≤d/dt|V_i_t - V_j_t|^2.

    Combining with (<ref>) and (<ref>), one obtains
    
    |d/dt𝒟_X| ≤𝒟_V,   d/dt𝒟_V ≤ -κ p/p-1ψ_0 𝒟_V,

    which is an SDDI. By Lemma <ref>, there exists a positive constant R_μ such that 
    
    𝒟_V(t)≤ 2e^-κψ_0 tR_μ.




    

  
Step 2.
    
    It can be obtained that 
    
    d/dt𝔼 | V_1 - V_1 |^q =  
               1/pd/dt𝔼∑_i=1^p (∑_k=1^d | V_i^k - V_i^k |^2)^q/2,

    with
    
    d/dt𝔼   (∑_k=1^d | V_i^k - V_i^k |^2)^q/2 
            =  𝔼[ q/2(∑_k=1^d | V_i^k - V_i^k |^2)^q/2-1d/dt∑_k=1^d 
    |V_i^k - V_i^k|^2 ]
    
            =    qκ/p-1𝔼[ |V_i - V_i|^q-2∑_j=1^p ψ(|X_j - X_i|)∑_k=1^d(V_j^k - V_i^k - V_j^k + V_i^k)( V_i^k - V_i^k). 
       .  + ∑_j=1^p 
            |V_i - V_i|^q-2(ψ(|X_j - X_i|) - ψ(|X_j - X_i|))
           ∑_k=1^d(V_j^k - V_i^k)( V_i^k - V_i^k) 
           ].

    Rewrite the right-hand side of (<ref>) as
   
    d/dt𝔼 | V_1 - V_1 |^q
           := I_21 + I_22,

   where
   
    I_21:= qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 (V_j^k - V_i^k - V_j^k + V_i^k)(V_i^k - V_i^k),

   and 
   
    I_22:= qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d
            (ψ(|X_j - X_i|) - ψ(|X_j - X_i|))
    ·|V_i - V_i|^q-2
           ( V_i^k - V_i^k )(V_j^k - V_i^k).

   First estimate I_21 using the index changing trick:
   
    I_21 =    qκ/p-1𝔼∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 [(V_j^k -V_j^k)-( V_i^k - V_i^k )](V_i^k - V_i^k)
    
               =    qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2 (V_j^k -V_j^k)(V_i^k - V_i^k).
       .- ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2(V_i^k - V_i^k)^2]
    
               =    qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_j - V_j|^q-2 (V_j^k -V_j^k)(V_i^k - V_i^k).
       .- ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) |V_i - V_i|^q-2(V_i^k - V_i^k)^2]
    
               =     -qκ/p-1𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|). 
       ·[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
               .(V_i^k - V_i^k)]
    
               =    qκ/2(p-1)𝔼[ ∑_i,j=1^p ∑_k=1^d  ψ(|X_j - X_i|) .
       .·[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)][(V_j^k - V_j^k)-(V_i^k - V_i^k)]].

    One can use the relation 
    
    ∑_k=1^d[|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
            ·.[(V_j^k - V_j^k)-(V_i^k - V_i^k)]] ≤ 0,

    and the condition (<ref>) of ψ to obtain
    
    I_21≤   qκ/2(p-1)ψ_0 𝔼{∑_i,j=1^p ∑_k=1^d 
            [|V_i - V_i|^q-2 (V_i^k -V_i^k) - |V_j - V_j|^q-2 (V_j^k -V_j^k)]
       · [(V_j^k - V_j^k)-(V_i^k - V_i^k)]}
    
            =    qκ/2(p-1)ψ_0 𝔼[∑_i=1^p ∑_k=1^d 
            |V_i - V_i|^q-2 (V_i^k -V_i^k)∑_j=1^p (V_j^k -V_j^k).
       -  ∑_i=1^p ∑_k=1^d 
             p|V_i - V_i|^q-2 (V_i^k -V_i^k)^2
            -  ∑_j=1^p ∑_k=1^d 
             p|V_j - V_j|^q-2 (V_j^k -V_j^k)^2
       +  .∑_j=1^p ∑_k=1^d 
             |V_j - V_j|^q-2 (V_j^k -V_j^k)∑_i=1^p (V_i^k -V_i^k)].

    By the momentum condition 𝔼∑_i=1^p (V_i^k - V_i^k) = 0, it holds
    
    I_21≤  -p^2qκ/p-1ψ_0 𝔼|V_1 - V_1|^q.


   For I_22, one has
    
    I_22≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p ∑_k=1^d | |X_j - X_i| - 
                |X_j - X_i | | V_i - V_i|^q-2|V_i^k - V_i^k| |V_j^k - V_i^k|
                ]
    ≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p| |X_j - X_i| - 
                |X_j - X_i | | |V_i - V_i|^q-1 |V_j - V_i|
                ],

    by the Cauchy inequality. Then by the inequality | |a-b| - |c-d| | ≤ |a-c| + |b-d|, one has
   
    I_22≤   qκ/p-1ψ_Lip𝔼[
                ∑_i,j=1^p| |X_i - X_i| + 
                |X_j - X_j | | |V_i - V_i|^q-1 |V_j - V_i|
                ]
    ≤   qκ/p-1ψ_Lip𝒟_V(t) 𝔼[p
                ∑_i=1^p |X_i - X_i| | V_i - V_i |^q-1 + 
                ∑_i,j=1^p |X_j - X_j| | V_i - V_i |^q-1]
    ≤   2q p^2κ/p-1ψ_Lip𝒟_V(t) 𝔼[
                 |X_1 - X_1| | V_1 - V_1 |^q-1],

    by Hölder's inequality. Now by combining (<ref>) and (<ref>) and (<ref>), one has
    
    d/dt𝔼| V_1 - V_1 |^q ≤    -p/p-1qκψ_0 
                𝔼|V_1 - V_1|^q 
       + 2p/p-1qκψ_Lip𝒟_V(t)
                𝔼[|X_1-X_1| |V_1 - V_1|^q-1]
    ≤    -p/p-1qκψ_0 
                𝔼|V_1 - V_1|^q 
       + 2p/p-1qκψ_Lip𝒟_V(t)
                (𝔼[|X_1-X_1|^q])^1/q (𝔼|V_1 - V_1|^q)^1-1/q,

    similarly by Hölder's inequality. Since
    
    d/dt (𝔼   |V_1 - V_1|^q)^1/q = 
                1/q(𝔼|V_1 - V_1|^q)^1/q-1d/dt𝔼|V_1 - V_1|^q
    ≤    -pκ/p-1ψ_0 
                (𝔼[|V_1 - V_1|^q_q])^1/q 
                + 2pκ/p-1ψ_Lip𝒟_V(t)
                (𝔼[|X_1-X_1|^q])^1/q.

    and 
    
    d/dt𝔼|X_1 - X_1|^q =    𝔼q/2(∑_k=1^d |X_1^k - X_1^k|^2)^q/2-1∑_k=1^d(X_1^k - X_1^k)· 2(V_1^k - V_1^k)
    ≤   𝔼 q |X_1 - X_1|^q-1 |V_1 - V_1 |
    ≤    q (𝔼|X_1 - X_1|^q)^1-1/q (𝔼|V_1 - V_1|^q)^1/q,

    thus it holds 
    
    d/dt(𝔼|X_1 - X_1|^q)^1/q = 
            1/q(𝔼|X_1 - X_1|^q)^1/q-1d/dt𝔼|X_1 - X_1|^q
            ≤ (𝔼|V_1 - V_1|^q)^1/q.

    Note that 𝒟_V≤ 2 e^-κψ_0 t  R_μ by Step 1. 
    Now one knows that both (𝔼|X_1 - X_1|^q)^1/q and (𝔼|V_1 - V_1|^q)^1/q satisfy Lemma <ref> and then there exists a constant C such that
    
    (𝔼|X_1 - X_1|^q)^1/q(t)
            + (𝔼|V_1 - V_1 ^q)^1/q(t)
            ≤ C [(𝔼|X_1 - X_1 |^q)^1/q(0)
            + (𝔼|V_1 - V_1 |^q)^1/q(0)].

    Then, 
    
    W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤   (𝔼 |X_1 -X_1|^q (t) + 𝔼 |V_1 -V_1|^q(t))^1/q
    ≤   (𝔼|X_1 - X_1|^q (t))^1/q
             + (𝔼|V_1 - V_1|^q (t))^1/q
    ≤    C [(𝔼|X_1 - X_1|^q)^1/q(0)
            + (𝔼|V_1 - V_1|^q)^1/q(0)]
    ≤   2 C [𝔼 |X_1 -X_1|^q (0) + 𝔼 |V_1 -V_1|^q(0)]^1/q,

    since a^1/q+ b^1/q≤ 2^1-1/q(a+b)^1/q for a≥0, b≥ 0 with q≥ 1. For any ϵ>0, the initial data can be chosen by a coupling such that 
    
    [𝔼 |X_1 -X_1|^q (0) + 𝔼 |V_1 -V_1|^q(0)]^1/q≤ϵ + W_1(μ_1,μ_2).

    Hence by the arbitrariness of ϵ, it holds W_q (𝒬_∞ (μ_1) ,𝒬_∞ (μ_2)) ≤ C W_q(μ_1,μ_2),







    
    For f defined in (<ref>) with R<∞, one has
    
    W_q(𝒬_∞^n-m+1(f_m-1),𝒬_∞^n-m(f_m))
            ≤ C W_q(𝒬_∞(f_m-1), f_m),    q=[1,∞),

    where C is relevant to ψ,κ,R and d.
    
    
        
        By the flocking property shown in Step 1 in Lemma <ref>, one has supp^v[𝒬_∞(f_m-1)]⊂supp^v[f_m-1]. Also, supp^v[f_m-1]⊂supp^v[f_0] by Lemma <ref>. Furthermore, for n-m-1≥ k≥ 0, 𝒬^k+1_∞ (f_m-1) and 𝒬^k_∞ (f_m) have compact supports, where 𝒬^0_∞ (f_m):=f_m.
        
        Similarly with the proof of Proposition <ref>, note that (X_1(kτ),V_1(kτ) and (X_1(kτ), V_1(kτ)) obey 𝒬^k+1_∞ (f_m-1) and 𝒬^k_∞ (f_m) respectively, which satisfy the momentum equality assumption of Lemma <ref>. 
    
        
        Denote 
        
    |Δ X_1|:=(𝔼|X_1 - X_1|^q)^1/q,   |Δ V_1|:=(𝔼|V_1 - V_1|^q)^1/q,   
            |Δ Z_1|:=|Δ X_1|+|Δ V_1|,

        for q≥ 1.
        Hence, by the analysis in Lemma <ref>, (<ref>) and (<ref>) still hold for t∈ [0,(n-m)τ].  
        By Lemma <ref>, one obtains
        
    |Δ Z_1 ((n-m)τ)| ≤ C |Δ Z_1 (0)|.

        By choosing a suitable coupling of the initial data, the proof is finished.
    


    

 §.§ An auxiliary system

    
    We introduce an auxiliary system (<ref>) with law S(f_t_k) to get the estimate on 
    W_q (𝒬_∞ (f_t_k), f_t_k+1) ≤ W_q(S(f_t_k),f_t_k+1) + W_q(S(f_t_k),𝒬_∞ (f_t_k) ).
 

    Rewrite (<ref>) into the following particle system
    
    {d/dtX_i(t) =   V_i(t),  i=1,⋯, N,
    d/dtV_i(t) =   ∫κψ(|x-X_i|)(v-V_i) f_t(x,v)dxdv.
            .


    In order to show the law of (X,V) at time nτ is close to 𝒬_∞^n, we consider the transient system
    
    {d/dtX̂_i(t) =   V̂_i(t),  i=1,⋯, N,
    d/dtV̂_i(t) =   ∫κψ(|x-X̂_i|)(v-V̂_i) f_t_k(x,v)dxdv,
            .

    with (X̂(0),V̂(0))∼f_t_k.
    Denote S(f_t_k):=Law(X̂(τ),V̂(τ)).

    
    Under the same assumption of Lemma <ref>, it holds
        
    supp[S(f_t_k)]⊂
                B(0,C)× B(0,C e^-κψ_0 t_k+1).

    
    
           For S(f_t_k), define
        
    𝖷_k (t) :=   max{ |x(t,x^k,v^k)|  | (x^k,v^k)∈supp[f_t_k]},
    𝖵_k (t) :=   max{ |v(t,x^k,v^k)|  | (x^k,v^k)∈supp[f_t_k]},

        where the pair (x(t,x^k,v^k), v(t,x^k,v^k)) denotes (x(t),v(t)) being the solution of (<ref>) with (x(t_k),v(t_k)) = (x^k,v^k). Note that the maximum is reached because it is assumed that supp[f_t_k] is compact. Denote K_t^x ⊂supp^x [f_t_k] (resp. K_t^v ⊂supp^v [f_t_k]) the set of points (x^k,v^k) such that the maximum is reached in 𝖷_k(t) (resp. in 𝖵_k(t)). By definition, one has |𝖷_k(t)|^2 = |x(t,x^k,v^k)|^2 for every (x^k,v^k)∈ K_t^x. It follows from Danskin's theorem <cit.> and from the fact that ∂_t x(t,x^k,v^k) = v(t,x^k,v^k) that
        
    𝖷_k(t)·𝖷_k(t) = max{ x(t,x^k,v^k) · v(t,x^k,v^k) | (x^k,v^k)∈ K_t^x }.

        Using the Cauchy-Schwarz inequality, one obtains that 
        
    𝖷_k(t) ≤𝖵_k(t).

        Similarly, by  Danskin's theorem again, one has
        
    𝖵_k(t)·𝖵_k(t) = max{ v(t,x^k,v^k) ·∫κψ(|x-x(t,x^k,v^k)|)(v-v(t,x^k,v^k)) f_t_kdxdv
    | (x^k,v^k)∈ K_t^v }.

        Note that by the definition of K_t^v, v(t,x^k,v^k) · (v-v(t,x^k,v^k))≤ 0. It holds that 
        
    𝖵_k(t)·𝖵_k(t)  ≤
                -κψ_0 ∫ v(t,x^k,v^k) · v(t,x^k,v^k) f_t_k dxdv,   (x^k,v^k)∈ K_t^v ,

        since ∫ v f_t_k dxdv = 0 and ∫f_t_kdxdv =1. Then one has 
        
    𝖵_k(t) ≤ -κψ_0 𝖵_k(t),

        and thus (<ref>) holds.
    


    
    Under the same assumption of Lemma <ref>, there exists a positive constant α such that
        
    W_q(S(f_t_k),f_t_k+1)≤ C e^-κψ_0 t_kτ^2,    q∈[1,∞),

        with C=C(κ,ψ_0,ψ_Lip, R).
    
    
    
    
    
    
        Let ℛ(x,v,t) = ∫ψ(|y-x|)(w-v)(f_t(y,w)-f_t_k(y,w))dydw. For t∈ [t_k,t_k+1), by straightforward calculus, it holds
        
        
    | ∂ℛ/∂ t (x,v,t)| =    | ∫ψ(|y-x|)(w-v) ∂f_t/∂ t(y,w)dydw|
    
                    =    | ∫ψ(|y-x|)(w-v) [ - w·∇
                    _y f_t(y,w) ..
       + ..∇_w ·(f_t(y,w)κ∫ψ (|y-y|)(w-w)f_t(y,w)dydw)]dydw |
    
                    =    | ∫[w·∇_y ψ(|y-x|)(w-v) f_t(y,w)..
        - . .dκψ(|y-x|)f_t(y,w)∫ψ (|y-y|)(w-w)f_t(y,w)dydw]dydw |
    ≤   ψ_Lip∫ |w| |v-w| f_t (y,w) dydw 
                    + dκ∫ |w-w| f_t(y,w) f_t (y,w) dydwdydw
    ≤   ψ_Lip∫ (|w| |v| + |w|^2) f_t (y,w) dydw 
                    + dκ∫ 2|w| f_t(y,w) dydw.

        
         By Lemma <ref>, 
         
    |∂ℛ/∂ t(X_1,V_1,t)| ≤ C e^-κψ_0 t.

        
        
        
        
        We first consider q to be an even integer.
        For (X_1,V_1) obeying (<ref>) and (X̂_1,V̂_1) obeying (<ref>), with (X_1,V_1) (t_k) = (X̂_1,V̂_1) (t_k) ∼f_t_k, it holds
        
    d/dt   |V_1 -V̂_1|^2 =  2∑_k=1^d(V^k_1 -V̂^k_1)d/dt(V^k_1 -V̂^k_1)
    
                    =    2κ∑_k=1^d(V^k_1 -V̂^k_1)
                     [∫ψ (|x-X_1|) (v - V_1) f_t_k(x,v) dxdv .
       .- ∫ψ (|x-X̂_1|) (v - V̂_1) f_t_k(x,v) dxdv + ℛ(X_1,V_1)]^(k)
    
                    =     2κ∑_k=1^d(V^k_1 -V̂^k_1)( ∫ψ (|x-X_1|) [(v - V_1)-(v-V̂_1)] f_t_k(x,v) dxdv)^(k)
        + 2κ∑_k=1^d(V^k_1 -V̂^k_1) (∫ [ψ (|x-X̂_1|)- ψ (|x-X_1|)](v - V̂_1) f_t_k(x,v) dxdv ) ^(k)
        + 2κ∑_k=1^d(V^k_1 -V̂^k_1) ℛ(X_1,V_1)^(k)
    ≤     -qψ_0 | V_1 - V̂_1|^2 
                    +2κψ_Lip   |X_1-X̂_1 | ∑_k=1^d
                    (V^k_1 -V̂^k_1)(∫ (v-V̂_1)f_t_k)^(k)
       + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,
    ≤     -2κψ_0 | V_1 - V̂_1|^2 
                    +2κψ_Lip   |X_1-X̂_1|  |V_1 - V̂_1 |
                    |V̂_1|
       + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,

        where the superscript (k) means the k-th dimension of vectors. Since ∫ vf_t dxdv = v =0 by (<ref>), and |V̂_1|≤ C e^-κψ_0 t_k by Lemma <ref> and Lemma <ref>, it holds
        
    d/dt |V_1 -V̂_1|^2   
                    ≤   -2κψ_0 | V_1 - V̂_1|^2 
                    +Cκψ_Lip   e^-κψ_0 t_k |X_1-X̂_1|  |V_1 - V̂_1 |
    
                    + 2κ |V_1 - V̂_1 | |ℛ(X_1,V_1)|,

        and
        
    d/dt|V_1 - V̂_1| ≤ -ψ_0 κ |V_1 - V̂_1|
                + Cκψ_Lip   e^-κψ_0 t_k |X_1-X̂_1|
                + κ  |ℛ(X_1,V_1)|.

        Also, with
        
    d/dt | X_1 - X̂_1| ≤ |V_1-V̂_1|,

        one has
        
    d/dt(|X_1-X̂_1| + |V_1-V̂_1| ) ≤ C(|X_1-X̂_1| + |V_1-V̂_1| ) + C|ℛ|.

        By Grönwall's inequality, it holds
        
    |X̂_1-X_1| + |V̂_1-V_1| ≤ C e^-κψ_0 t_kτ^2.

        Then
        
    [𝔼[|X̂_1-X_1|^q + |V̂_1-V_1|^q]]^1/q≤[𝔼[|X̂_1-X_1| + |V̂_1-V_1|]^q]^1/q≤ C e^-κψ_0 t_kτ^2,

        which implies
        
    W_q(S(f_t_k),f_t_k+1)≤ C(κ,ψ_0,ψ_Lip, R ) e^-κψ_0 t_kτ^2.

    



    
    
    
    The estimate on W_q(S(f_t_k),𝒬_∞ (f_t_k) ) is given by Lemma <ref>.
    
    
    
    
        Under the same assumption of Theorem <ref>, it holds
        
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ, R) τ^2,    q∈ [1,∞).

    
    
    
    Set (X̂_1,V̂_1) and {(X_i,V_i)} to be the solutions of (<ref>) and (<ref>) respectively. The initial data (X̂_1(0),V̂_1(0)) and {(X_i(0),V_i(0))} follow f_t_k. Take the coupling (X̂_1(0),V̂_1(0)) = (X_1(0),V_1(0)). We introduce {(X_i^E, V_i^E)} as the approximation of {(X_i,V_i)} by the forward Euler method with time step τ. Then, taking the expectation of the selection of {(X_j(0),V_j(0)}_j 1, one has 
    
    d/dt1/2𝔼 |V_1^E - V̂_1|^2 =   ∫κψ (| x- X_1(0)|) (v- V_1(0))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
        - ∫κψ (| x- X̂_1(t)|) (v- V̂_1(t))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
    ≤   κ∫ψ (|x-X_1(0)|) (V̂_1(t)-V_1(0))·𝔼(V_1^E(t)-V̂_1(t))f_t_k
       + κ∫ψ_Lip |X_1(0)-X̂_1(t)| (v-V̂_1(t))·𝔼(V_1^E(t)-V̂_1(t))f_t_k.

    By Lemma <ref> and Lemma <ref>, one has 
    
    | v-V̂_1(t)|≤ Ce^-κψ_0 t_k.

    Note that 
    
    |X_1(0)-X̂_1(t)|≤ Ce^-κψ_0 t_kτ

    and 
    
    |V̂_1(t)-V_1(0)|≤ Ce^-κψ_0 t_kτ.

    Take the expectation of (X_1(0),V_1(0)), one has
    
    d/dt𝔼 |V_1^E - V̂_1|^2
             ≤ Ce^-κψ_0 t_kτ√(𝔼|V_1^E - V̂_1|^2).

    Hence
    
    𝔼 |V_1^E - V̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^4.

    Similar with (<ref>), one has
    
    𝔼 |X_1^E - X̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^6.

    Now consider
    
    d/dt1/2𝔼|V^E_1(t) -V_1(t)|^2 =   𝔼(V^E_1(t) -V_1(t))κ/p-1∑_j=1^p[ψ(X_j(0)-X_1(0))(V_j(0)-V_1(0)) 
       - ψ(X_j(t) - X_1(t))(V_j(t) - V_1(t))]
    ≤   𝔼κ(V^E_1 -V_1)[ψ_Lip|X_2(0)-X_1(0)- X_2(t) + X_1(t))(V_2(0)-V_1(0)) 
       + ψ(X_2(t) - X_1(t))(V_2(0) - V_1(0) - V_2(t) + V_1(t))].

    By Lemma <ref> and Lemma <ref>, one knows that |V_j(t)|≤ e^-κψ_0 t_k, for any j and t∈ [0,τ]. Hence it's easy to obtain  
    
    d/dt𝔼 |V_1^E - V_1|^2
             ≤ Ce^-κψ_0 t_kτ√(𝔼|V_1^E - V_1|^2),

    and 
    
    𝔼 |V_1^E - V_1|^2 ≤ Ce^-2κψ_0 t_kτ^4,   𝔼 |X_1^E - X̂_1|^2 ≤ Ce^-2κψ_0 t_kτ^6.

    Then, by the triangular inequality of Wasserstein distance, one has
    
    W_2 (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ, R)e^-κψ_0 t_kτ^2.

    The estimate of W_q is similar by calculating 
    
    d/dt(𝔼|V_1^E - V̂_1|^q)^1/q=1/q(𝔼|V_1^E - V̂_1|^q)^1/q-1d/dt𝔼|V_1^E - V̂_1|^q ≤  C(κ, R)e^-κψ_0 t_kτ.

        
    
    


    

 §.§ Proof of Theorem <ref>

    Now we prove Theorem <ref> by combining the above lemmas.
    
    
    By the triangular inequality, it holds
    
    W_q(𝒬_∞^n (f_0), f(nτ))  ≤∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m)).

    And then, by the stability result in Lemma <ref>, it holds
    
    ∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m))
            ≤∑_m=1^n C W_q(𝒬_∞ (f_m-1), f_m).

    Again by the triangular inequality, one has
    
    W_q(𝒬_∞ (f_m-1), f_m) ≤ W_q(𝒬_∞ (f_m-1), S(f_m-1))
            + W_q(S(f_m-1), f_m).

    Then, by Lemma <ref> and Lemma <ref>, we have
    
    ∑_m=1^n W_q (𝒬_∞^n-m+1 (f_m-1),𝒬_∞^n-m(f_m))
            ≤∑_m=1^n  C e^-κψ_0 (m-1)ττ^2 ≤  C τ,

    where the constant C is independent of n.
        One completes the proof.
    
    
    
    
        Although our analysis is based on W_q,2 distance, the main results in Section <ref> also work on W_q,p, p∈ [1,∞), by noting that
    
    x _p≤ d^1-1/q+1/px _q,    x∈ℝ^d,    1<p,q<∞.

    







§ APPLICATION: THE RBM-GPC


In this section, we show a simple application for the mean-field limit analysis for the Cucker-Smale model combined with the RBM. 

Based on the above analysis, and inspired by Carrillo's work <cit.>, we can construct the numerical scheme for the approximation of stochastic mean-field models of swarming which preserve the nonnegativity of the solution. 

Let us consider the stochastic mean-field equation for the distribution function f(θ, x, v; t):

    ∂_t f + v·∇_x f = ∇_v · [ℋ[f]f],

where

    ℋ[f](θ,x,v;t) = κ∫∫ψ(|x-y|,θ)(v-w)f(θ,y,w;t)dwdy,

with ψ satisfying  positivity, boundedness, Lipschitz continuity and mononticity conditions: there exists a positive constant ψ_M  > 0  such that 

    0≤ψ(r,θ) ≤ψ_M, ∀ r≥0,   ψ_Lip<∞,


    (ψ(r_1,θ)-ψ(r_2,θ))(r_1-r_2)≤ 0,   r_1,r_2 ∈ℝ_+.

Here, θ is a random variable from a measurable space (Ω,ℱ) to (I_Θ,ℬ_ℝ), with a probability distribution π (θ), with I_Θ⊂ℝ and ℬ_ℝ the Borel set.

It is easy to see that the total mass and momentum of (<ref>) is conserved in time.
Note that if we directly apply the classical stochastic Galerkin method to (<ref>), the solution might lose its positivity. However, we can make use of the microscopic dynamics to approximate f. Our method is to consider the corresponding Cucker-Smale model with random input θ on the interaction kernel ψ:

    {∂_t x_i(t,θ) =    v_i(t,θ),  i=1,⋯, N,
    ∂_t v_i(t,θ) =   κ/N-1∑_j=1^N 
            ψ(|x_j(t,θ)-x_i(t,θ)|,θ)(v_j(t,θ)-v_i(t,θ)).
        .


Denote {Φ_k(θ)}_j=0^∞ to be the gPC basis with respect to π, which means that Φ_k is a polynomial of degree k, and
these polynomials form an orthonormal basis of L^2(Ω,ℱ,π). 
We approximate the position and the velocity of i-th agent as follows:

    x_i(θ,t)≈ x^K_i = ∑_k=0^K^kx̂_i(t)Φ_k(θ),
           v_i(θ,t)≈ v^K_i = ∑_k=0^K^kv̂_i(t)Φ_k(θ),

where

    ^kx̂_i=∫ x_i(θ,t)Φ_k(θ)dπ,   ^kv̂_i=∫ v_i(θ,t)Φ_k(θ)dπ.

We obtain the following system

    {d/dt^ℓx̂_i (t) =   ^ℓv̂_i (t),
    d/dt^ℓv̂_i (t) =   κ/N-1∑_j=1^N∑_k=0^K e_ℓ k^ij(^kv̂_j(t)-^kv̂_i(t)),
           l = 0,1,…,K..

where

    e_ℓ k^ij=1/|Φ_ℓ|^2∫ψ(|x_j-x_i|,θ) Φ_k(θ)Φ_ℓ(θ)dπ.

Then, we design Algorithm <ref>, on the generalized Polynomial Chaos expansion in the random space based on the RBM:




    Although being derived from different numerical methods (the RBM and the Monte Carlo method), Algorithm <ref> is similar with the Monte Carlo gPC (MCgPC) proposed in <cit.> at first glance. Both can reduce the total cost of (<ref>) to 𝒪(KN). The main difference between them is that the RBM-gPC preserves the momentum during evolution. Actually, since the RBM updates information in the unit of batches, (instead, the MCgPC updates information of each particle independently at per time step.) for any t∈[t_m,t_m+1], the total sum of velocities is conserved in each batch by calculus:
    
    d/dt∑_i∈𝒞_q^ℓv̂_i (t) = κ/p-1∑_i,j∈𝒞_q∑_k=0^K e_ℓ k^ij(^kv̂_j(t)-^kv̂_i(t))=0,

    for l=0,⋯,K. Furthermore, it holds 
    
    ∑_i=1^N ^ℓv̂_i (t) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t_m) = ∑_q=1^N/p∑_i∈𝒞_q^ℓv̂_i (t_0) = ∑_i=1^N ^ℓv̂_i (0).

    










    We use 
    x_i^K,R(t,θ):= ∑_k=0^K^kx̂_i(t)Φ_k(θ),   v_i^K,R(t,θ):=∑_k=0^K^kv̂_i(t)Φ_k(θ)
 to denote the approximation of position and velocity obtained by Algorithm <ref>. The kinetic distribution f^R_N,K(θ,x,v,t) is recovered from the empirical density function 
    f^R_N,K(θ,x,v,t) = 1/N∑_i=1^N δ(x-x_i^K,R(t,θ))⊗δ(v-v_i^K,R(t,θ)).
 Common approximations methods for Dirac delta distributions can be used when needed. For example, we fix an upper and lower bound for the (x,v)-space, to restrict our computational domain in a cube space and discretize it in uniform cells {C_l}_l=1^N_l of size |C_l|=h^2d, where h is the edge length of each cell. One can reconstruct the distribution f^R_N,K by histograms counting the number of particles belonging to each cell. Then the discrete distribution function f^R_N,K reads
    
    f^R_N,K(θ,x,v,t) = 1/|C_l|(1/N∑_i=1^N 𝕀(x_i^K,R(t,θ),v_i^K,R(t,θ)∈ C_l))·(∑_l=1^N_l𝕀((x,v)∈ C_l)),

    where 𝕀(·) is the indicator function.
    Clearly, the positivity of the distribution function is preserved. One can also refer to a brief review, by Lee et al. <cit.>, on the numerical methods for regularized Dirac delta functions.








    With respect to the stochastic variable θ, the dynamics of the N-particle system achieves spectral accuracy as we adopt the generalized polynomial chaos (gPC) basis <cit.>, provided that we have a smooth dependence of the particle solution from the random field.

    For any given θ and K, denote the distribution function of the N-particle system (<ref>) by f_N,K. 
    
    If one uses the reconstruction (<ref>), it holds that
        
    𝔼 W_2 (f^R_N,K,f^R_N,K) ≤(𝔼1/N∑_i=1^N |(x_i^R,v_i^R) - c_l(i)|^2 )^1/2,

        where c_l(i) is the center of the cell C_l containing (x_i^R,v_i^R). Since h is the uniform edge length, |(x_i^R,v_i^R) - c_l(i)|^2 ≤∑^d (h/2)^2 for any i. Combining with the estimate of W_2 (f_N,K,f^R_N,K) in Section <ref>, it holds that
    
    
    
    
        
    𝔼 W_2 (f_N,K,f^R_N,K)     ≤𝒪(√(τ/p-1)) +
           {   𝒪(N^-1/4)     if  d<2,
       𝒪(N^-1/4√(log(1+N)))     if  d=2,
       𝒪(N^-1/d)     if  d>2,
            .
            + 𝒪(√(d)h).







 §.§ Numerical experiments

Here we present some numerical tests for the RBM-gPC. In the following, we take the Legendre gPC base.



  
The space homogeneous case
We consider the space-independent case in the one-dimensional setting:

    ∂_t f(θ,v;t) = ∂_v [ K(θ)(v-u)f(θ,v;t)],

with u = ∫ vf(v)dv and K(θ)=0.5 + 0.01θ, θ∼𝒰(0,1). The long time solution is a Dirac delta δ(v-u) provided K(θ)>0, see <cit.>. We take the Legendre polynomial basis. The gPC approximation is given for h=0,⋯,M by 

    ∂_t f̂_h (v;t) = 1/|Φ_h|^2∂_v [ ∑_k=0^M (v-u) ℋ_hkf̂_t_k(v;t)],

where

    ℋ_hk = ∫ K(θ)Φ_h (θ) Φ_k (θ) d g (θ),   f̂_h (v;t) = ∫ f(θ,v;t) Φ_h (θ) d g (θ).

We consider an initial density function f_0(v) by

    f_0(v) = β[exp(-(v-μ)^2/2σ^2) + exp(-(v+μ)^2/2σ^2)],   σ^2=0.1,  μ = 0.5,

with β>0 a normalization constant. Discrete samples of the initial data of the ODE system are obtained from f_0(v). Denote the expected temperature of the system 𝒯 by 

    𝒯(f) = ∫ (v-u)^2 f(θ,v;t)dvdg(θ).

In Figure <ref>, we show the evolution of the expected density toward the Dirac delta function δ(v-u), u=0 for the RBM-gPC scheme at different times with p=2 and N=10^4.

In Figure <ref>, we take T=0.5, K=3, dt=10^-2 and dv=10^-2. Time integration is performed through a 4th order Runge-Kutta method. In Figure <ref>, we evaluate the mean square error (MSE) for the expected temperature 

    MSE_T := 1/n_m∑_i=1^n_m|𝒯(f_ref) - 𝒯(f_rbm^(i)) |^2,

as N grows with p=2, where 𝒯(f_ref) denotes the reference expected temperature obtained by (<ref>) through a central difference scheme, and 𝒯(f_rbm^(i)) denotes the expected temperature obtained by the RBM-gPC in the i-th turn. We have applied n_m=100 for random initialization and batch splits. 
Figure <ref> shows the difference of TV distance between the expected reference distribution and the expected distribution by the RBM-gPC:

    Err_TV := | 𝔼_θ f_ref - 𝔼_θ1/n_m∑_i=1^n_m f_rbm^(i)|_L^1,

with n_m =100 as well.












Also, Figure <ref> and Figure <ref> show the decay of MSE_T as the batch size p and the time step dt, respectively. The reference distribution of Figure <ref> is obtained by the RBM-gPC with N=2^8, p = 2^8, and dt = 10^-2, and of Figure <ref> is obtained with N=2^8, p = 2, and dt = 10^-2. We still take n_m=100.






  
Stochastic 1D Cucker-Smale dynamics
In this test, we consider the 1D Cucker-Smale dynamics. Let us consider the initial distribution as the following bivariate and bimodal distribution

    f_0(x,v) = 1/2πσ_xσ_vexp(-x^2/2σ_v ^2)[exp(-(v+1)^2/2σ_x ^2)+ exp(-(v-1)^2/2σ_x^2)],

with σ_x^2 =0.5, σ_v^2=0.2. Our initial data for particle positions and velocities are
sampled from f_0. The stochastic interactions are given by

    H(θ; |x_i - x_j|) = 1/(1+|x_i - x_j|^2)^γ(θ),

with γ (θ) = 0.1+ 0.05 θ, θ∼𝒰[-1,1]. The results are obtained through the RBM-gPC scheme with dt = 10^-2, N=10^4, batch size p =2, and K=3. The expected distribution is reconstructed by  in Matlab in the domain [-3,3]×[-3,3] with 100 gridpoints in both space and velocity. Figure <ref> presents the evolution over the time interval t∈[0,4] of the expected distribution. 






  
Stochastic 2D Cucker-Smale dynamics
Figure <ref> shows the evolution over the time interval t∈ [0,4] of the 2D Cucker-Smale nonhomogeneous mean-field model. For the initial distribution, we consider uniformly distributed N=10^4 particles on a 2D annulus with a circular counterclockwise motion

    f_0 (x,v) = 1/|C|χ (x∈ C)δ( v - k∧ x/|x|),

being C: = {x∈ℝ^2: 0.5≤ |x_1-x_2|≤ 1 } and k the fundamental unit vector of the z-axis, where |C| means the cardinality of set C.
The interaction kernel shares the same form with (<ref>).
The evolution shows the flocking phenomenon in the top row and that the mean velocity is conserved in the bottom row of Figure <ref>. The results are obtained through the RBM-gPC scheme with dt = 10^-2, N=10^4, batch size p =2, and K=3. The reconstruction follows the same method as in the 1D case.








§ CONCLUSION

In conclusion, we give an analysis on the mean-field limit of the RBM for the Cucker-Smale model and obtain a long-time approximation. In addition, as a simple application, we propose a generalized Polynomial Chaos based Random Batch Method (RBM-gPC) as a numerical method to simulate the mean-field limit dynamics of the Cucker-Smale model with random inputs.

This paper focuses on the separate limits of the number of particles N and the time step τ but not a uniform limit of (N,τ). For future work, it will be interesting to investigate the mixture of particles as time evolves, and to study on the uniform limit. Also, numerically, an error estimate on this model applied with the particle method (corresponding to a new model) can be an interesting topic to study.







§ ACKNOWLEDGEMENT

We thank Prof. Shi Jin and Prof. Lei Li for detailed and valuable comments and advice on the manuscript. This work is supported by the NSFC grant (No. 12031013, 12201404), the Shanghai Municipal Science and Technology Key Project (No. 22JC1402300), and Project supported by the National Science Foundation for International Senior Scientists (No. 12350710181).



    
    

§ PROOF OF LEMMA <REF>

    The proof is given by Lemma 2.4 in <cit.>.
    
        For the first assertion, we claim that the relative velocities are non-increasing in time. Let t∈ [t_m-1, t_m) be given. Then one can choose time-dependent indices k and ℓ such that 
        
    |V_k^R (t) - V_ℓ^R (t)| = 𝒟_V^R(t).

        Then for such k and ℓ, one has 
        
    d/dt | V_k^R(t) -V_ℓ^R(t)|^2 =    
                2(V_k^R(t) - V_ℓ^R (t)) ·d/dt(V_k^R(t) - V_ℓ^R (t))
    
                =    2κ/p-1∑_j∈ [k]_mψ (|X_j^R - X_k^R|)(V_j^R - V_k^R)·(V_k^R - V_ℓ^R)
       - 2κ/p-1∑_j∈[i]_mψ (|X_j^R - X_ℓ^R|)(V_j^R - V_ℓ^R)·(V_k^R - V_ℓ^R).

        In order to show that the right-hand side of (<ref>) is not positive, we use the maximality of | V_k^R(t) -V_ℓ^R(t)| at time t. Since
        
    | V_k^R(t) -V_ℓ^R(t)| ≥ | V_j^R(t) -V_ℓ^R(t)|,    j = 1,⋯,N,

        one has
        
    (V_j^R(t)-V_k^R (t))·(V_k^R(t)-V_ℓ^R (t)) =
                - ((V_k^R-V_ℓ^R) - (V_j^R-V_ℓ^R))· (V_k^R - V_ℓ^R)
    ≤ - | V_k^R(t) -V_ℓ^R(t)|^2 + | V_j^R(t) -V_ℓ^R(t)|| V_k^R(t) -V_ℓ^R(t)|≤ 0,    j= 1,⋯,N.

        Similarly, one has 
        
    (V_j^R(t)-V_ℓ^R (t))· (V_k^R(t)-V_ℓ^R (t)) ≥ 0,    j=1,⋯,N.

        Hence, | V_k^R(t) -V_ℓ^R(t)|^2 is non-increasing in time.

        Then by the definition od 𝒟_X^R and 𝒟_V^R, one has 
        
    d/dt𝒟_X^R(t)≤𝒟_V^R(t) ≤𝒟_V^R(0),    a.e.  t>0.

        This yields the second assertion on 𝒟_X^R.
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        
    
    
    
    
    
    
    
    
    
    
    
    


    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


    




    
    
    
    
    

§ DETAILS ON PROPOSITION <REF>


    

  
Fixed-time regularity of f. First we check the regularity of f by Lemma <ref>, which is needed in the proof of Proposition <ref>.
    
    
    For the flocking kinetic model (<ref>), suppose that the initial probability measure f_0 ∈ (C^2∩ W^2,∞)(ℝ^2d) is compactly supported, and the interaction kernel ψ satisfies 
    sup_r≥ 0 (|ψ^'(r)| + |ψ^''(r)|) ≤ϵ_ψ
 with a constant ϵ_ψ. Then, for any T∈ (0,∞), there exists a unique classical solution f_t ∈ C^2([0,T)×ℝ^2d) for (<ref>). 

    The proof stem of Lemma <ref> is generally a smoothness check for f inspired by in [<cit.>, Theorem 3.1]. The main difference with <cit.> is that here we need better regularity for the operator (ℒ^*)^2 and (ℒ_k^*)^2 in Lemma <ref>.
    
    Local existence follows from [<cit.> Theorem 3.1] by the standard fixed point argument. One only needs to obtain a C^2 bound of f to conclude the global existence of classical solutions. One can express the non-conservative kinetic model (<ref>) in terms of the non-linear transport operator 𝒯:=∂_t + v·∇_x + ξ [f] ·∇_v,
    
    𝒯f = -f∇_v ·ξ[f].

    We claim that there exists a positive constant C(d,κ,ϵ_ψ, R) such that
    
    | 𝒯f| ≤ C |f|,

    
    | 𝒯(∂_x_if) | ≤ C (|f| + |∇_v f| + |∂_x_if)|,

    
    | 𝒯(∂_v_i f) | ≤ C (|∇_v f| + |∂_x_if|),

    
    | 𝒯(∂_x_i x_jf) | ≤ C (  |∂_x_if| + |∂_x_jf| + |∂_x_i x_jf| +|∇_v f| + |∇_v∂_x_if| + |∇_v∂_x_jf|),

    
    | 𝒯(∂_x_i v_jf) | ≤ C ( |∂_x_i x_jf| +|∇_v f| + |∂_v_j∂_x_if| + |∇_v∂_v_jf| + |∇_v∂_x_if|),


    
    | 𝒯(∂_v_i v_jf) | ≤ C ( |∂_x_i v_jf| + |∂_x_j∂_v_if| +  |∂_v_j∂_v_if|).


    
    To verify these inequalities, note that -∇_v·ξ [f] = κ d ∫ψ (|y-x|)f(y,w,t)dydw and thus |∇_v·ξ [f]| ≤κ d. By direct calculus, one obtains the following estimates:
 

    For (<ref>),
    
    | 𝒯f| ≤| -κf∇_v ·∫ψ(|y-x|) (w-v) f(y,w,t) dydw| ≤κ d |f|;

    for (<ref>),
    
    | 𝒯(∂_x_if) |    ≤| - (∂ _x_iξ [f]) ·∇_v f -  (∂_x_i∇_v ·ξ [f])f  -  (∇_v ·ξ [f])∂ _x_if|
       ≤κϵ_ψ (√(M_2^v f_0) + 2R) |∇_v f| + κ d ϵ_ψ |f| + κ d |∂_x_if|)
       ≤ C (|f| + |∇_v f| + |∂_x_if)|;

    For (<ref>),
    
    |𝒯∂_v_if | ≤ |-∂_x_if| + κ |∇_v f| + κ d |∂_v_if|
            ≤ C (|∇_v f| + |∂_x_if|).

    The estimate on (<ref>)-(<ref>) is similar and we omit it for brevity.
    
    Now, let F(t) measure the W^2,∞-norm of f,
    
    F := f(·,t) _W^2,∞ .

    The above inequalities imply
    
    d/dtF(t) ≤ CF(t).

    Then we end up with the energy bound by Grönwall's inequality
    F(t)≤ F(0) e^Ct.
    Equipped with this a priori W^2,∞ estimate, standard continuation principle yields a global extension of local classical solutions. 




The following is a simple prerequisite for Proposition <ref>.

    
For any g_1∈ L^1(ℝ^2d),g_2∈ L^1(ℝ^2dp), the operators ℒ_k^* and ℒ^* defined by (<ref>) and (<ref>) respectively, it holds
    
    ∫ | e^τℒ_k^*g_1| dz ≤∫ | g_1| dz,

    
    ∫|  e^τℒ^*  g_2  | dz_1 ⋯ dz_p ≤∫|  g_2  | dz_1 ⋯ dz_p.


     
    
    One only needs to consider g_1≥ 0 since there is composition g_1 = g_1^+ -g_1^-. Also, since e^τℒ_k^*cg_1 = ce^τℒ_k^*g_1 and e^τℒ^*cg_2 = ce^τℒ^*g_2 for any constant c, one can set g_1 be a probability density function. Then,
    
    ∫ |e^τℒ_k^*g_1| dz = ∫ e^τℒ_k^*g_1 dz,

    since e^τℒ_k^*g_1 is another probability density function derived from g_1 by the dynamics (<ref>). It's clear that
    
    ∫ e^τℒ_k^*g_1 dz = 
        ∫∫_0^τ∂/∂ tg_1(t)dtdz + ∫ g_1 dz = ∫ g_1dz,

    where g_1(t) denotes the solution of (<ref>) with initial datum g_1. For the general g_1, 
    
    ∫ | e^τℒ_k^*g_1| dz ≤∫ | e^τℒ_k^*g_1^+| dz
             + ∫ | e^τℒ_k^*g_1^-| dz
             =  ∫ | g_1^+| dz + ∫ | g_1^-| dz =  ∫ | g_1| dz.

    The proof of (<ref>) is similar.





    
    

  
Proof of Proposition <ref>.
    It's clear that S = e^τℒ_k^*f_t_k, where 
    
    ℒ_k^* := -  (v ·∇_x) · -
        ∇_v ·( ∫κψ(|y-x|)(w-v)f_t_k(y,w)dydw ·).

    Define the operator of (<ref>) as
    
    ℒ^* := -∑_i=1^p (v_i∇_x_i)·  - ∑_i=1^p∇_v_i· (ξ_i ·).

    Then one has
    
    𝒬_∞ (f_t_k) =   ∫ e^τℒ^*Π_i=1^p f_t_k(x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p 
    
            =   f_t_k (x_1,v_1) + τ∫ℒ^* Π_i=1^p f_t_k (x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p 
       + ∫_0^τ (τ -s) ∫ (ℒ^*)^2 e^sℒ^*Π_i=1^p f_t_k (x_i,v_i)dx_2⋯ dx_pdv_2⋯ dv_p ds,

    where
    
    
    ∫ℒ^* Π_i=1^p f_t_k (z_i) dz_2⋯ dz_p 
                =     -∑_i=1^p ∫ (v_i ∇_x_i)·Π_j=1^p f_t_k (z_j) - ∇ _v_i· (ξ_i Π_j=1^p)f_t_k (z_j)dz_2⋯ dz_p)
    
                =     - (v_1 ∇_x_1) ·f_t_k (z_1) - ∇_v_1·∫κψ (|y-x_1|)(w-v_1) f_t_k (y,w) dydw.

    
    Note that S(f_t_k) = f_t_k + τℒ_k^*f_t_k + ∫_0^τ (τ -s)(ℒ_k^*)^2 e^sℒ_k^*f_t_k ds, then
    
        
    | S(f_t_k) - 𝒬_∞ (f_t_k) |(z_1)≤∫ _0^τ (τ-s) [ |(ℒ_k^*)^2 e^sℒ_k^*f_t_k(z_1)|  +  | ∫ (ℒ^*)^2 e^sℒ^*Π_i=1^p f_t_k (z_i) dz_2⋯ dz_p | ]ds.

    
    
    As shown in Lemma <ref>, f_t_k∈ C^2(z), hence it holds 
    
    ∫ |(ℒ_k^*)^2 f_t_k| dz≤ C ,  ∫| (ℒ^*)^2 Π _i=1^p f_t_k (z_i)| dz_2 ⋯ dz_p≤ C,
 
    where C=C(κ,ϵ_ψ ,R,d). By Lemma <ref>, taking g_1 = (ℒ_k^*)^2  f_t_k and g_2 = (ℒ^*)^2 Π _i=1^p f_t_k (z_i), one has
    
    ∫ |(ℒ_k^*)^2 e^τℒ_k^*f_t_k| dz ≤∫ |(ℒ_k^*)^2 f_t_k| dz,

    and
    
    ∫| (ℒ^*)^2 e^τℒ^*Π _i=1^p f_t_k (z_i)| dz_1 ⋯ dz_p ≤∫| (ℒ^*)^2 Π _i=1^p f_t_k (z_i)| dz_1 ⋯ dz_p,

    Using the exchangability of ℒ^* and e^τℒ^*, with ℒ^*=ℒ_k^* or ℒ^*. 
    
    Then one has |S(f_t_k) - 𝒬_∞ (f_t_k)|_TV≤ C τ^2. 
    
    For S(f_t_k), by Lemma <ref>, it holds
    
    supp[S_∞(f_t_k)]⊂
            B(0,C)× B(0,C e^-κψ_0 t).


    For the support of 𝒬_∞(f_t_k), by Lemma <ref>, one knows that 
    
    supp[𝒬_∞(f_t_k)]⊂
            B(0,C)× B(0,C e^-κψ_0 t_k).

    
    Hence for any z ∈supp|S(f_t_k) - f_t_k|, it holds |z|≤ C.
    Then, by Lemma <ref> and applying similar analysis on Lemma <ref> Step 1, one has 
    
    W_q (𝒬_∞ (f_t_k),S(f_t_k)) ≤ C(κ,ϵ_ψ , f_0,d) τ^2/q.















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