Paper Title: Bounding adapted Wasserstein metrics.













 
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Bounding adapted Wasserstein metrics
    Jose Blanchet, Martin Larsson, Jonghwa Park, Johannes Wiesel
    August 5, 2024
================================================================



§ ABSTRACT 
The Wasserstein distance 𝒲_p is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance 𝒜𝒲_p extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems.
    
    While the topological differences between 𝒜𝒲_p and 𝒲_p are well understood, their differences as metrics remain largely unexplored beyond the trivial bound 𝒲_p≲𝒜𝒲_p. This paper closes this gap by providing upper bounds of 𝒜𝒲_p in terms of 𝒲_p through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of 𝒲_p, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on 𝒲_p automatically hold for 𝒜𝒲_p under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality 𝒜𝒲_1≤ C√(𝒲_1) on the set of measures that have Lipschitz kernels. 

    Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.

    
    

    Keywords: (adapted) Wassestein distance, optimal transport, modulus of continuity, relative compactness




Introduction

For N∈ we denote by _p(^N) the set of all Borel probability measures on ^N that have finite p-moments, where 1≤ p<∞ is fixed throughout this paper. The Wasserstein distance, defined via 

    _p(μ, ν)
        :=(inf_π∈(μ, ν)∫_^N×^Nx-y^p π(dx, dy))^1/p,

is a metric on _p(^N) that metrizes weak convergence plus convergence of p-moments. Here · denotes the Euclidean norm on ^N and (μ, ν) is the set of all couplings between μ∈_p(^N) and ν∈_p(^N), i.e., those probability measures on ^N×^N whose first marginal is μ and whose second marginal is ν. Computing the Wasserstein distance is an instance of the Kantorovich optimal transport problem <cit.>, and its numerous mathematical properties are well studied. We refer to <cit.> for a general overview.

While the Wasserstein distance between laws on ^N has seen a surge of applications in the last few years, the situation changes  when μ,ν are interpreted as laws of discrete-time stochastic processes in their natural filtrations. In this case, measuring the distance between μ and ν via _p(μ, ν) is often inadequate when considering time-dependent optimization problems such as optimal stopping problems and multistage optimization. The key reason for this is that the formulation (<ref>) does not take the time-structure of the laws μ,ν into account when optimizing over all possible couplings on ^N. The restriction to so-called bicausal or adapted couplings described in the next paragraph addresses this issue.

Let us first  set up some notation: consider two stochastic processes X=(X_1, …, X_T) and Y=(Y_1,…, Y_T) in finite discrete time, where both X_t and Y_t are ^d-valued random variables for t∈{1,2,…, T}. Throughout the paper, T∈ denotes the number of time steps and d∈ is the dimension of the state space. Consider now a Monge map S=(S_1, …, S_T):(^d)^T→ (^d)^T from the law of X to the law of Y, which satisfies

    Y_td= S_t(X_1, …, X_T)    for  t∈{1,2,…, T}.

Without further restrictions on the map S, Y_t can depend on the whole stochastic process X. In other words, it is impossible to  determine Y_t only from the information (X_1, …, X_t) as the map S_t uses information about the evolution of X after time t in general. In order to respect the filtration generated by X, it is reasonable to require S to be adapted, i.e., S_t should only be a function of the first t coordinates. As shown e.g. in <cit.>, the notion of adapted mappings is generalized to bicausal couplings as follows.


Let μ and ν be two probability measures on (^d)^T. A coupling π∈(μ, ν) is called bicausal if for (X,Y)∼π and t∈{1,2,…, T-1},

    (Y_1, …, Y_t)  and  (X_t+1, …, X_T)  are conditionally independent given  X_1, …, X_t

and

    (X_1, …, X_t)  and  (Y_t+1, …, Y_T)  are conditionally independent given  Y_1, …, Y_t.

The set of all bicausal couplings between μ and ν is denoted by (μ, ν). 


Heuristically speaking, π∈(μ, ν) couples two processes X∼μ and Y∼ν such that predicting Y at time t requires at most the information of X up to time t and vice versa. We define the adapted Wasserstein distance as the optimal transport cost over bicausal couplings.
[The adapted Wassertein distance]
    Let μ, ν∈_p((^d)^T). The adapted Wasserstein distance between μ and ν is given by
    
    _p(μ, ν)
            :=(inf_π∈(μ, ν)∫_(^d)^T× (^d)^T∑_t=1^Tx_t-y_t^p π(dx, dy))^1/p.



Much of the previous work has been on characterizing the topology generated by _p; we provide a literature review of these results in Section <ref>. Here we only remark that by Definition <ref> one can easily derive the inequality _p≤ C_p (for some constant C>0 depending only on p, T). As a consequence, the adapted Wasserstein topology is finer than the Wasserstein topology. The following example going back at least to <cit.> shows that this inclusion is strict in general. We refer to it as the standard example and use it to illustrate our results throughout this paper.

[Standard example]
    Let μ_=1/2δ_(, 1)+1/2δ_(-, -1) and μ=1/2δ_(0,1)+1/2δ_(0,-1). As → 0 we have _p(μ, μ_)=→ 0, but _p(μ, μ_)=(^p+2^p-1)^1/p↛ 0. 






In fact, the adapted Wasserstein topology is the coarsest topology that makes optimal stopping problems continuous <cit.>. <cit.> demonstrate, that time-dependent optimization problems from mathematical finance such as utility maximization, pricing and hedging derivatives, risk measurements, etc., are actually Lipschitz continuous with respect to _p.

While the differences between the Wasserstein topology and the adapted Wasserstein topology are thus well understood, the comparison between the metrics _p and _p is still largely open to the best of our knowledge. This article aims to fill this gap. We want to answer the following question:


Can we identify a class of subsets K⊆_p((^d)^T), on which _p and _p are nearly equivalent, i.e. there are “nice" explicit upper bounds of _p in terms of _p?


Except for trivial bounds, the above is an ill-posed task for K=_p((^d)^T). In fact, Example <ref> clearly shows that _p and _p cannot be equivalent in any meaningful sense. However, this example is arguably tailor made to showcase the differences between _p and _p. In fact, {μ_:>0} are chosen in such a way that the conditional distributions x_1↦(X^_2∈ · |X^_1=x_1) become increasingly irregular for (X^_1, X^_2)∼μ_: recall that we have

    (X^_2= 1|X^_1=ϵ)=1  and  (X^_2=-1|X^_1=-ϵ)=1.

In other words, {μ_: >0} does not have equicontinuous kernels. In many situations it is however reasonable to restrict to subsets  K⊆_p((^d)^T), whose elements have reasonably smooth kernels. It turns out that such subsets K are exactly the _p-relatively compact sets.



A set K⊆_p((^d)^T) is _p-relatively compact if and only if K is _p-relatively compact and

    lim_δ→ 0sup_μ∈ Kω^t,p_μ(δ)=0
     for all  t∈{1,2,…, T-1},

where ω^t,p_μ:(0,∞)→ (0,∞) is the p-modulus of continuity of μ defined via

    ω_μ^t,p(δ)
                :=sup{([_p((X_t+1∈ ·  X_1, …, X_t), (Y_t+1∈ ·  Y_1, …, Y_t))^p])^1/p}.

Here, the supremum is taken over all X,Y∼μ such that (X_1, …, X_t)-(Y_1, …, Y_t)_L^p<δ.


Let us emphasize here that focusing on the _p-relatively compact subsets of _p((^d)^T) is usually not a big restriction in practical applications, and sometimes even arises naturally. We refer to Section <ref> for a more detailed discussion.

Proposition <ref> will prove central in this article, as we will essentially show that the inequality _p≤ C_p can be reversed exactly on the _p-relatively compact sets. To get some intuition for this, assume for now that p=1 and K⊆_1((^d)^T) is _1-compact. Then it essentially follows from <cit.>, that the Wasserstein topology and the adapted Wasserstein topology agree on K. As a consequence, for any σ>0 there exists a constant C depending on σ and K such that we have

    _1(μ,ν)≤ C_1(μ,ν) +σ

for all μ, ν∈ K. While theoretically appealing, this result is of very limited practical relevance as long as one cannot characterize the dependence of C on σ and K. One of the main contributions of this article is to make the bound (<ref>) explicit. In fact, we show that (ignoring dimensional constants that can be made explicit) for any positive real numbers 0<σ<R, 

    _1(μ,ν)
    ≲R/σ_1(μ,ν) + h(σ,K) + g(R,K),

where the function h is derived from Eder's modulus of continuity ω (see Proposition <ref>) and the function g characterizes the tail behavior of measures in K outside a Euclidean ball of radius R. In particular, lim_σ→ 0 h(σ,K)=0 and lim_R→∞ g(R,K) =0. We refer to Corollary <ref> for an exact statement of this bound.

In other words, (<ref>) implies that _1 can be bounded by _1 up to an error that is uniformly bounded on _1-relatively compact sets. This seemingly simple result shows its true power in applications: the time-dependent optimization problems (e.g. utility maximization, pricing, hedging) mentioned above are not just continuous in _1, but are actually continuous in _1 on _1-relatively compact sets and estimates are explicitly computable. This significantly simplifies the analysis of time-dependent optimization problems. Indeed, it is actually not necessary to design new estimates specifically tailored for _1 as e.g. done in <cit.>; instead one can simply rely on the rich theory that has already been developed for _1 in the last decades. We give some examples illustrating this new methodology below, showing how one can derive finite-sample guarantees for the adapted Wasserstein distance. We believe that  (<ref>) has much further-reaching implications for the theory of adapted transports however, which we aim to investigate in future work.



 §.§ The smooth adapted Wasserstein distance


In order to derive (<ref>), we first need to introduce a variant  of _p: 


Given a noise distribution ξ and a bandwidth parameter σ>0, we define the smooth adapted Wasserstein distance via 
    ^(σ)_p(μ, ν):=_p(μ∗ξ_σ, ν∗ξ_σ).

Here ∗ denotes the convolution of two measures and ξ_σ is the law of (x↦σ x)_#ξ, i.e. if Z∼ξ, then σ Z∼ξ_σ.


 The non-adapted counterpart of ^(σ)_p is called the smooth Wasserstein distance 
    ^(σ)_p(μ, ν):=_p(μ∗ξ_σ, ν∗ξ_σ)
 and is by now well studied in the statistics and machine learning literature, mainly for the purpose of relaxing the curse of dimensionality. Indeed, denoting the empirical measure of μ by μ_n:=1/n∑_j=1^nδ_X^(j) where X^(1), X^(2), … are i.i.d samples of μ, it turns out that √(n)^(σ)_p(μ, μ_n) has a weak limit under suitable moment assumptions. We refer to <cit.> and the references therein for this line of research. On the other hand, ^(σ)_p is quite new to the best of our knowledge, and has only been studied very recently in the literature, cf. <cit.>. These results suggest that, as in the non-adapted case, the empirical convergence rate only exhibits a mild dependence on the dimension after smoothing.

While the statistical properties of ^(σ)_p(μ_n, μ) are thus at least partially understood, the arguably more fundamental question of how smoothing affects the adapted Wasserstein topology is still wide open to the best of our knowledge. Recall that it is well known that _p and ^(σ)_p generate the same topology (see <cit.>), and it is thus natural to conjecture that the same holds for _p and ^(σ)_p. Perhaps rather surprisingly, we show that this conjecture is false in general. In fact we prove that, essentially,

    Wasserstein topology ⊆ smooth adapted Wasserstein topology ⊆ adapted Wasserstein topology,

and the inclusions in the above display are equalities under suitable assumptions on the decay of σ (when thinking about sequential convergence). We return to this discussion in Section <ref>.



 §.§ TEXT
 

We are now in a position to give  a heuristic derivation of (<ref>). To keep the notation simple, we take p=1, T=2 and refer to Corollary <ref> for the general statement. Assuming K⊆_1((^d)^2) is _1-relatively compact, an application of the triangle inequality yields

    _1(μ, ν)
        ≤^(σ)_1(μ, ν)
        +2sup_μ∈ K_1(μ, μ∗ξ_σ)

for any μ, ν∈ K and σ>0. In order to bound _1, it is thus sufficient to bound the two terms on the right-hand side of (<ref>) separately. These bounds constitute the two main results of our paper and we refer the reader to Section <ref> for exact statements. We start with a bound on the smooth adapted Wasserstein distance.

 
For any σ, R>0 we have

    ^(σ)_1(μ, ν)
        ≲R/σ_1(μ, ν)
        +sup_μ∈ K∫_{x≥ R}x (μ∗ξ_σ)(dx).


In other words, ^(σ)_1 can be bounded by _1 up to an error term depending only on the tails of μ∗ξ_σ. In the next section we show that the scaling of order _1/σ in the upper bound is essentially optimal. Our second main result pertains to the question, how much information is lost by considering μ∗ξ_σ instead of its unperturbed version μ. 


We have

    sup_μ∈ K_1(μ, μ∗ξ_σ)
        ≲σ+sup_μ∈ Kω^1,1_μ(σ).



In conclusion, the difference between μ and its smoothed version μ∗ξ_σ is bounded by Eder's modulus of continuity ω^1,1_μ uniformly on the _1-relatively compact sets.

Combining Theorems <ref> and <ref> with  (<ref>) finally yields the following result:


For all 0<σ<R we have

    _1(μ, ν)
        ≲R/σ_1(μ, ν)
        +(σ+sup_μ∈ Kω^1,1_μ(σ))
        +sup_μ∈ K∫_{x≥ R}xμ(dx).



Returning to (<ref>), we conclude that we can choose

    h(σ, K)
        =σ+sup_μ∈ Kω^1,1_μ(σ),  
        g(R, K)
        =sup_μ∈ K∫_{x≥ R}xμ(dx).


To summarize, the functions h,g are remarkably explicit and exactly mirror Eder's characterization of _1-relatively compact sets as stated in Proposition <ref>. In particular, g controls _1-relative compactness of K, while h controls the regularity of the kernels in K. 
The proof of the general result stated in Corollary <ref> essentially proceeds in the same fashion as described above, iterating over time periods t=1, …, T-1.




 §.§ Comparison of Wasserstein topologies 
 

To the best of our knowledge, general properties of the topology generated by _p^(σ) have not been studied. Given our explicit estimates in Theorems <ref> and <ref>, we can now address this question in great generality. To make our results concise, we consider a sequence (μ_n)_n∈ in _p((^d)^T) and an additional measure μ∈_p((^d)^T). We  refer to Section <ref> for exact statements.

Perhaps surprisingly, for fixed σ>0, the topology generated by ^(σ)_p is in fact the (vanilla) Wasserstein topology, i.e. 

    lim_n→∞^(σ)_p(μ,μ_n) =0   ⇔  lim_n→∞^(σ)_p(μ,μ_n) =0   ⇔  lim_n→∞_p(μ,μ_n) =0.

In this sense, all adapted topologies are equal as famously stated in <cit.>, but only if there is no uncertainty about the observed distributions μ, μ_n. As soon as these distributions are perturbed by some (arbitrarily small) independent noise ξ_σ, there is virtually no difference between the Wasserstein topology and the adapted Wasserstein topology. Phrased differently, in this case the finer structure of the adapted Wasserstein topology is absorbed by the noise ξ_σ.

The picture is different when one considers variable noise (ξ_σ_n)_n∈ instead. Now there is an interesting tradeoff between the size of the noise ξ_σ_n and the convergence speed of _1(μ,μ_n) on the one hand (cf. Theorem <ref>) and the regularity of kernels on the other hand (cf. Theorem <ref>). Figure <ref> illustrates this tradeoff in full generality. Here, the function h^t,p_μ_n is an iterated modulus of continuity, e.g., h^1,p_μ_n(σ_n)=ω^1,p_μ_n(σ_n), h^2,p_μ_n(σ_n)=ω^2,p_μ_n(σ_n+h^1,p_μ_n(σ_n)) and h^3,p_μ_n(σ_n)=ω^3,p_μ_n(σ_n+h^1,p_μ_n(σ_n)+h^2,p_μ_n(σ_n)). See Theorem <ref> for the precise definition of h^t,p_μ_n.






In the sense of Figure <ref>, convergence in ^(σ_n)_p is in general neither the same as convergence in _p nor convergence in _p and the smoothed adapted Wasserstein distance ^(σ_n)_p exhibits a non-trivial interpolation property between the different topologies. 
To motivate the above results, it is perhaps most instructive to return to the standard Example <ref>, where we can compute all relevant quantities explicitly. By convolving μ__n=1/2δ_(_n, 1)+1/2δ_(-_n, -1) with Gaussian noise ξ_σ_n=(0, σ^2_n_2), we see from Figure <ref> that kernels of μ__n are smoothed out. As _n→ 0, we can show that

    
  * _n/σ_n→ 0 if and only if _1(μ,μ__n)/σ_n→ 0 if and only if ^(σ_n)_p( μ, μ__n)→ 0, 
    
  * _n/σ_n→∞ if and only if ω^1,p_μ__n(σ_n)→ 0 if and only if _p(μ__n, μ__n∗ξ_σ_n)→ 0;

see Appendix <ref> for details. Roughly speaking, if σ_n decays slowly enough (so that _n/σ_n→ 0), the behavior of ^(σ_n)_p(μ, μ__n) mirrors the one of _p(μ, μ__n). Similarly, if σ_n decays fast enough (so that _n/σ_n→∞), we can use the triangle inequality to see that ^(σ_n)_p(μ, μ__n) and _p(μ, μ__n) converge toward the same value. In this sense, the tradeoff illustrated in Figure <ref> captures exactly the right scaling of _p(μ, μ__n) and σ_n, at least for the standard example.









 §.§ Related work
 

Adapted variants of the Wasserstein distance have been studied independently by many authors. In <cit.>, Aldous introduced the notion of extended weak convergence, which is essentially induced by the joint law between the original process and its corresponding prediction process. Further derivations of adapted topologies can be found in <cit.> and the references therein. Remarkably, <cit.> shows that all these seemingly different approaches define the same topology, which we refer to as the adapted Wasserstein topology throughout this paper. Based on the information topology proposed by <cit.>, Eder  characterizes the relatively compact sets in the adapted Wasserstein topology in <cit.>. The modulus of continuity proposed by Eder plays a central role in our work.

As discussed earlier, adapted transport maps induce bicausal couplings and as for their non-adapted counterparts, this inclusion is dense under mild regularity assumptions on the kernels <cit.>. The key idea is to embed bicausal couplings into enlarged spaces and approximate them by adapted maps. In Lemma <ref> we adapt these results to approximate a bicausal coupling between μ and μ∗ξ_σ by transport maps in order to bound the adapted Wasserstein distance between them. 

We also mention that duality of the adapted Wasserstein distance is investigated in <cit.>. Furthermore, <cit.> introduce a dynamic programming principle for the adapted Wasserstein distance, which we will use throughout this paper.

Notably, Eckstein–Pammer show in <cit.>, that the total variation distance and the adapted total variation distance (μ, ν):=inf_π∈(μ, ν)π({x≠ y}) are equivalent as metrics. To the best of our knowledge, this is the first result that gives upper and lower bounds of an adapted optimal transport cost by an optimal transport cost. Our work uses this estimate to bound _p in terms of _p.

Regarding applications, <cit.> show that many multistage optimization problems such as computing bid prices, are Hölder continuous with respect to _p. <cit.> extend this idea to a continuous time setting in mathematical finance and show that utility maximization, risk measurement and pricing/hedging problems are Lipschitz continuous with respect to _1.

Smoothing measures by convolving with Gaussian noise ξ is a standard tool in statistics, and is often used to obtain dimension-free empirical approximations. In optimal transport, the smooth Wasserstein distance and its concentration has been studied extensively.
We refer e.g. to <cit.> for both the slow rate [^(σ)_p(μ, μ_n)^p]≤ C/√(n) and the fast rate [^(σ)_p(μ, μ_n)]≤ C/√(n) under suitable moment conditions. Distributional limits of √(n)^(σ)_p(μ, μ_n) can be found in <cit.> for the case p=1 and in <cit.> for p>1.

It is well-known that the usual empirical measure μ_n is not adequate for empirical approximations in the adapted Wasserstein topology, i.e., lim_n→∞_p(μ,μ_n)≠ 0 in general. To alleviate this, <cit.> considers so-called adapted empirical measures for measures supported on compact sets, and prove the same dimension-dependent convergence rates as are known for their non-adapted counterparts. <cit.> extends this result to general measures. 

Quite recently, there have been  first efforts to study properties of the smooth adapted Wasserstein distance ^(σ)_p. Motivated by a martingale pair test, <cit.> obtain the  dimension-free convergence rate 1/√(n) for the so-called smoothed martingale projection distance. Under suitable moment assumptions, <cit.> establishes rates for [^(σ)_1(μ, μ_n)].

Organization of the paper
The rest of the paper is organized as follows. We end this introduction by setting up notation in Section <ref>.  In Section <ref> we present essential background materials for this paper, including the characterization of _p-relatively compact sets by <cit.> and basic properties of smooth Wasserstein distances. In Section <ref>, we provide the exact statements of the results that were briefly mentioned in Section <ref> and Section <ref>. Proofs of results in Section <ref> are contained in Section <ref>. Section <ref> is devoted to proofs of results in Section <ref>. In Appendix <ref>, the convergence results regarding Example <ref> are explained in detail. Appendix <ref> contains a denseness result used in the proof of Lemma <ref>.


Notation
As mentioned above, · is the Euclidean norm, and we denote the scalar (dot) product by ·. Throughout the paper,  T∈ is the number of time steps and d∈ is the dimension of the state space. The set of all Borel probability measures on (^d)^T  is denoted by ((^d)^T). For 1≤ p<∞, _p((^d)^T) is the set of all μ∈((^d)^T) that have finite p-moments. If μ∈((^d)^T), then we write M_p(μ):=∫_(^d)^Tx^p μ(dx) for the p-moment of μ. We denote the pushforward measure of μ under a Borel function g by g_#μ, i.e. g_#μ(A)=μ({x:  g(x)∈ A}) for all Borel sets A.

For x∈ (^d)^T, we use the shorthand notation x_t to denote the t-th coordinate of x and x_1:t:=(x_1, …, x_t) for t∈{1,2,…, T}. In particular, x=x_1:T. Analogously, for μ∈((^d)^T) and t∈{1,2,…, T}, the measure μ_t∈(^d) is the projection of μ onto the t-th coordinate, i.e., if P_t:(^d)^T→^d is defined as P_t(x_1, …, x_T)=x_t, then μ_t=(P_t)_#μ. Also, μ_1:t∈((^d)^t) is the pushforward measure of μ through the projection map P_1:t onto the first t coordinates, i.e., P_1:t(x_1, …, x_T)=(x_1, …, x_t). When integrating with respect to μ_t(dx_t) or μ_1:t(dx_1:t), we often drop subscripts and write

    ∫_^dg(x_t) μ_t(dx_t)
        =∫_^dg(x_t)μ(dx_t),  ∫_(^d)^tg(x_1:t) μ_1:t(dx_1:t)
        =∫_(^d)^tg(x_1:t)μ(dx_1:t)

if there is no confusion. Given x∈ (^d)^T we define μ_x_1:t∈(^d), t∈{1,2,…, T} via

    μ_x_1:t(dx_t+1)
        =(X_t+1∈ dx_t+1 X_1:t=x_1:t)

for X∼μ∈((^d)^T). 
In other words, (μ_x_1:t)_x_1:t∈ (^d)^t are the disintegrations of μ_1:t+1 along the first t coordinates.

Recall that (μ, ν) is the set of all couplings, (μ, ν) is the set of all bicausal couplings, the Wasserstein distance _p is defined in (<ref>) and _p in Definition <ref>. We adopt similar shorthand notation as above for π∈((^d)^T× (^d)^T): π_t,t∈(^d×^d) (respectively, π_1:t, 1:t∈((^d)^t× (^d)^t)) is the pushforward measure of π through the map (x_1, …, x_T, y_1, …, y_T)↦ (x_t, y_t) (respectively, (x_1, …, x_T, y_1, …, y_T)↦ (x_1, …, x_t, y_1, …, y_t)). We use the abbreviations π(dx_t, dy_t)=π_t,t(dx_t, dy_t) and π(dx_1:t, dy_1:t)=π_1:t, 1:t(dx_1:t, dy_1:t) as in (<ref>) if there is no confusion. Let us also remark that by definition we have

    π_x_1:t, y_1:t(dx_t+1, dy_t+1)
        =(X_t+1∈ dx_t+1, Y_t+1∈ dy_t+1 X_1:t=x_1:t, Y_1:t=y_1:t)

for (X, Y)∼π. 

Next we introduce the convolution operation ∗, i.e., given two probability measures μ, ν, we define μ∗ν(A):=∫μ(A-x)ν(dx) for all measurable A. In particular, if ν has density g, then μ∗ν has density μ∗ g(x):=∫ g(x-y)μ(dy). The (distributional) Fourier transform of μ∈((^d)^T), which we denote by μ: (^d)^T→, is given by μ(t)=∫_(^d)^T e^-2π i t· xμ(dx). For a function g, we similarly have g(t):=∫_(^d)^T e^-2π i t· xg(x)dx. Recall that μ∗ν=μν.

Unless otherwise stated, we reserve the notation ξ for the smoothing noise distribution. For σ>0, we write ξ_σ for the law of σ Z, where Z∼ξ. We use the shorthand notation μ^σ:=μ∗ξ_σ. See Section <ref> for details.

For any signed measure ρ on (^d)^T we define ρ_:=sup∑_j=1^∞ρ(E_j), where the supremum is taken over all measurable partitions (E_j)_j∈ of (^d)^T. Recall that if μ, ν∈((^d)^T),

    μ-ν_
        =2sup_G⊆ (^d)^T( μ(G)-ν(G))
        =2sup_π∈(μ, ν)π({x≠ y}).


Given μ∈^N and Σ∈^N× N, we write (μ, Σ) for the Gaussian distribution with mean vector μ and covariance matrix Σ. The N× N identity matrix is denoted as _N. In particular, (0, _N) is the standard Gaussian measure on ^N. 

We write C for a constant, that might change from line to line. We will detail the dependence of C on non-trivial parameters throughout the paper.







§ BACKGROUND

In this section, we review some auxiliary results on characterizations of relative compactness in Wasserstein distance and smooth Wasserstein distances. Proofs of all results in this section can be found in Section <ref>.



 §.§ Compactness in the adapted Wasserstein topology

Recall that _p was introduced in (<ref>) and _p was introduced in Definition <ref>. Given x≤ C (∑_t=1^T x_t^p)^1/p, where C=1 if 1≤ p≤ 2 and C=T^1/2-1/p if p≥ 2, we have _p≤ C_p. Hence, the _p-relatively compact subsets of _p((^d)^T) are also _p-relatively compact. In fact, Eder showed in <cit.> that the _p-relatively compact sets are exactly those _p-relatively compact sets whose modulus of continuity converges uniformly.

    [Modulus of continuity, <cit.>]
        Let μ∈_p((^d)^T). The p-modulus of continuity of μ, denoted by ω_μ^t,p:(0,∞)→ [0, ∞), t∈{1,2,…, T-1}, is given by
        
    ω_μ^t,p(δ)
                =sup{([_p(μ_X_1:t, μ_Y_1:t)^p])^1/p |  X, Y∼μ, ([X_1:t-Y_1:t^p])^1/p<δ}.

    

<ref>
    
        Let K⊆_p((^d)^T). Then K is _p-relatively compact if and only if K is _p-relatively compact and
        
    lim_δ→ 0sup_μ∈ Kω^t,p_μ(δ)=0
                   for all  t∈{1,2,…, T-1}.

    
   thm-1
The characterization of compactness in Proposition <ref> is slightly different from the original result by Eder <cit.>. However, it is not hard to see that they are equivalent, as we prove in Section <ref>.
    
        Trivially K={μ} is _p-relatively compact. Hence, ω^t,p_μ(δ)→ 0 as δ→ 0.
    
    
        Recall μ_=1/2δ_(-, -1)+1/2δ_(, 1) from Example <ref>. Note that K:={μ_: 0<<1} is _p-relatively compact. However, (μ_)_x=δ_1 if x= and (μ_)_x=δ_-1 if x=-. Thus we easily compute that ω_μ_^1,p(δ)=(δ/)∧ 2 and sup_0<<1ω_μ_^1,p(δ)=2. Proposition <ref> implies that K is not _p-relatively compact.
     
Next we present some basic properties of the modulus of continuity ω.
    
        Let μ∈_p((^d)^T) and t∈{1,2,…, T-1}.
        
            
  *  The function ω^t,p_μ is monotone increasing, i.e. ω^t,p_μ(δ_1)≤ω^t,p_μ(δ_2) for δ_1<δ_2.
            
  *  For k, δ>0, ω^t,p_μ(kδ)≤ (k∨ 1)ω^t,p_μ(δ).
        
    
    If the kernels (^d)^t∋ x_1:t↦μ_x_1:t∈_p(^d) are equicontinuous,  where _p(^d) is endowed with a metric _p, then the modulus of continuity of μ converges uniformly. An important class of meausres with equicontinuous kernels is given by the ones with  Hölder continuous kernels.
    
        Let L>0 and 0<α≤ 1. The set _p, α, L((^d)^T) contains all μ∈_p((^d)^T) that satisfy
        
    _p(μ_x_1:t, μ_y_1:t)≤ L x_1:t-y_1:t^α

        for all x_1:t, y_1:t∈ (^d)^t and t∈{1, 2, …, T-1}.
    

    
        Let L>0 and 0<α≤ 1. If μ∈_p, α, L((^d)^T), then ω^t,p_μ(δ)≤ L δ^α for δ>0 and t∈{1,2,…, T-1}.
    
   
What is more, measures with Lipschitz kernels (i.e. α=1) arise in many practical examples. We refer to <cit.> for some simple sufficient conditions.
    




 §.§ Smooth adapted Wasserstein distances

As explained in Section <ref>, we define the smooth (adapted) Wasserstein distance as follows.
    [Smooth distances]
        Let ξ∈_p((^d)^T). For σ>0 and Z∼ξ, denote by ξ_σ the law of σ Z. We define the smooth p-Wasserstein distance with the smoothing parameter σ via
        
    ^(σ)_p(μ, ν)
                :=_p(μ∗ξ_σ, ν∗ξ_σ)  for μ, ν∈_p((^d)^T).

        Similarly, the smooth p-adapted Wasserstein distance with the smoothing parameter σ is given by
        
    ^(σ)_p(μ, ν)
                :=_p(μ∗ξ_σ, ν∗ξ_σ)  for μ, ν∈_p((^d)^T).

    
    
        For μ∈((^d)^T), we often use the shorthand notation μ^σ=μ∗ξ_σ. In particular, ^(σ)_p(μ, ν)=_p(μ^σ, ν^σ) and ^(σ)_p(μ, ν)=_p(μ^σ, ν^σ).
    
    In the following proposition, we present some properties of the smooth Wasserstein distance ^(σ)_p. While these results are simple extensions of the classical theory, we provide proofs in Section <ref> to keep our work self-contained.
    
    
        Suppose that the set of real zeros of ξ has Lebesgue measure zero. Then, for σ>0, ^(σ)_p is a metric on _p((^d)^T) that induces the same topology as _p. Furthermore, if σ_1, σ_2>0, then
        
    |^(σ_1)_p(μ, ν)-^(σ_2)_p(μ, ν)|
                ≤ 2σ_1-σ_2(M_p(ξ))^1/p

        for μ, ν∈_p((^d)^T). Here, recall that M_p(ξ)=∫_^dTx^p ξ(dx).
    
    The assumption that ξ does not admit any real zeros a.s. is essential to guarantee that ^(σ)_p defines a metric. This is satisfied, for example, when ξ is compactly supported (see <cit.>) or when ξ is Gaussian. The same result is true for ^(σ)_p.
    
    
        Suppose that the set of real zeros of ξ has Lebesgue measure zero. Then _p^(σ) is a metric on _p((^d)^T) for σ>0.
    




§ MAIN RESULTS

Now, we have all the necessary ingredients to present the main results. Proofs for this section can be found in Section <ref>. We begin with some standing assumptions on the  noise measure ξ. 
    
 We have the following:
        
            
  *  The noise distribution ξ∈_p((^d)^T)  has density f:(^d)^T→ which is continuously differentiable and f∈ L^1((^d)^T; (^d)^T). In particular, for σ>0, we write  f_σ(·):=σ^-dTf(·/σ) for the density of ξ_σ.
            
  *  The set of real zeros of f (or equivalently ξ) has Lebesgue measure zero.
        
    
    Assumption <ref> is a mild growth condition on f. Note from Theorem <ref> that <ref> guarantees that ^(σ)_p (as well as ^(σ)_p) is indeed a metric on the space of probability measures.

    

 §.§ The adapted Wasserstein distance and the total variation norm

     Recall that the p-Wasserstein distance _p(μ, ν) can be controlled by the p-total variation norm, ·^p(μ-ν)_. For example, <cit.> shows that
     
    _p(μ, ν)
             ≤ 2^1-1/p(∫x^p μ-ν(dx))^1/p.

     Similarly, Proposition <ref> shows that _p is bounded by the total variation norm ·_ plus remainder terms that can be made arbitrarily small. This result is based on <cit.> stating that the metric (μ, ν)=inf_π∈(μ, ν)π({x≠ y}) is equivalent to the total variation distance 1/2μ-ν_=inf_π∈(μ, ν)π({x≠ y}).
    
    Let μ, ν∈_p((^d)^T). For any R>0 we have
        
    _p(μ, ν)^p
                ≤ 6^p-1T2^T (R^p μ-ν_
                +∫_{x≥ R}x^p (μ+ν)(dx)).

    
    Our first main result, Theorem <ref>, asserts that the smooth p-adapted Wasserstein distance ^(σ)_p can be bounded by the 1-Wasserstein distance _1. 
    
    Let μ, ν∈_p((^d)^T). For any R>0 and σ>0 we have
        
    ^(σ)_p(μ, ν)^p
                ≤ 6^p-1T2^T (R^p f_L^1_1(μ, ν)/σ
                +∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)).

    
    Under additional assumptions on the measures μ and ν, we can actually drop the additive terms characterizing the tail behavior.

    
        For any σ>0 we have the following:
        
            
  *  If μ, ν, ξ∈_q((^d)^T) for some q>p, then
            
    ^(σ)_p(μ, ν)^p
                    ≤ 6^p-1T2^T+1(M_q(μ^σ+ν^σ))^p/q( f_L^1_1(μ, ν)/σ)^1-p/q.

            
  *  If μ, ν and ξ have bounded supports, then
            
    ^(σ)_p(μ, ν)^p
                     ≤ 6^p-1T2^T+2(((μ)∪(ν))+σ((ξ)))^p/σ f_L^1_1(μ, ν).

            
  *  Suppose ξ∼(0, _dT). If μ, ν∈_q((^d)^T) for some q>p, then for 0<σ_0<σ,
            
    ^(σ)_p(μ, ν)^p
                    ≤ 6^p-1T2^T+1 f^1-p/q_L^1(M_q(μ^σ+ν^σ))^p/q(^(σ_0)_1(μ, ν)/√(σ^2-σ^2_0))^1-p/q.

        
    

    Combining Theorem <ref> with the slow rate of ^(σ)_1 yields the following upper bound on the convergence rate between the true and empirical measure in ^(σ)_p. Similar bounds were announced in <cit.>.

     Let μ∈_q((^d)^T) for some q>p∨ (dT+2). Suppose ξ∼(0, _dT). Denote by μ_n the empirical measure of μ with sample size n∈, i.e. μ_n=1/n∑_j=1^n δ_X^(j) where X^(1), X^(2), …, X^(n) are i.i.d samples of μ. Then
        
    [^(σ)_p(μ, μ_n)^p]
                ≤ C(1+σ^(q+p)/2/σ^(1+dT/2)(1-p/q)) n^-(q-p)/(2q)

        for some constant C>0 that depends only on d, T, p, q, M_q(μ).
    

    
        If μ has a bounded support, then we have
        
    [^(σ)_p(μ, μ_n)^p]≤ C n^-1/2+

        for all >0. This recovers the slow rate [^(σ)_p(μ, μ_n)^p]≤ Cn^-1/2 up to an -loss.
    
   

    

 §.§ Smoothing and the modulus of continuity

    Our second main result pertains to the bandwidth effect of μ^σ. As shown in <cit.>, the smoothing operation is stable in the sense that _p(μ, μ^σ)→ 0 as σ→ 0. Here, we establish that convergence is controlled by the modulus of continuity defined in Definition <ref>.
        
     Let μ∈_p((^d)^T). Then for any σ>0,
        
    _p(μ, μ^σ)
                ≤ T(1∨(M_p(ξ))^1/p)∑_t=0^T-1h^t,p_μ(σ)

        where h^t,p_μ is defined as follows
        
    h^0,p_μ(σ)=σ,    
                h^t,p_μ(σ)=ω_μ^t,p(∑_s=0^t-1 h^s,p_μ(σ))  for  t∈{1,2, …, T-1}.

    

    
         The following hold.
        
            
  *  For L>0 and 0<α≤ 1, there exists a constant C>0 that depends only on T, (M_p(ξ))^1/p, L, α such that
            
    sup_μ∈_p, α, L((^d)^T)_p(μ, μ^σ)
                    ≤ C ∑_t=0^T-1σ^α^t.

            
  *  Let K⊆_p((^d)^T). If sup_μ∈ Kω^t,p_μ(δ)→ 0 as δ→ 0 for all t∈{1,2,…, T-1}, then
            
    lim_σ↘ 0sup_μ∈ K_p(μ, μ^σ)=0.

            In particular,
            
    lim_σ↘ 0sup_μ, ν∈ K^(σ)_p(μ, ν)-_p(μ, ν)
                    =0.

        
    

    By choosing a suitably decaying sequence (σ_n)_n∈, ((μ_n)^σ_n)_n∈ approaches μ in _p. The convergence rate depends on the dimension dT.
    
    
        Let μ∈_q((^d)^T)∩_p, α, L((^d)^T) for some q>p∨ (dT+2), L>0 and 0<α≤ 1. Suppose ξ∼(0, _dT). Using the same notation as in Corollary <ref>, there exists a constant C>0 that depends only on d, T, p, q, M_q(μ), L, α such that
        
    [_p(μ, (μ_n)^σ_n)^p]
                ≤ C n^-β

        where
        
    σ_n = n^-1-p/q/(dT+2)(1-p/q)+2pα^T-1 and β = pα^T-1(1-p/q)/(dT+2)(1-p/q)+2pα^T-1.

    


    

 §.§ TEXT

    Now, we turn our attention to bounding _p in terms of _1. By introducing an auxiliary noise ξ satisfying Assumption <ref>, a straightforward application of the triangle inequality to Theorem <ref> and Theorem <ref> shows the following proposition.
    
        Let μ, ν∈_p((^d)^T). Then for any R>0, σ>0 and ξ satisfying Assumption <ref> we have
        
    _p(μ, ν)^p
                   ≤ 18^p-1T2^T R^p f_L^1_1(μ, ν)/σ
                +18^p-1T2^T∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)
       +3^p-1T^p(1∨ M_p(ξ))(∑_t=0^T-1h^t,p_μ(σ))^p
                +3^p-1T^p(1∨ M_p(ξ))(∑_t=0^T-1h^t,p_ν(σ))^p.

    
    We remark that R>0, σ>0 and ξ (or equivalently f) can be chosen arbitrarily in the above, as long as Assumption <ref> is satisfied. We present several estimates as outcomes of specific choices of R, σ or ξ.

    
        Let K⊆_p((^d)^T) be _p-relatively compact. Then there exists a constant C>0 that depends only on d, T, p such that for all 0<σ<R and μ, ν∈ K,
        
    _p(μ, ν)^p
                ≤ C(R^p _1(μ, ν)/σ
                +sup_γ∈ K(∑_t=0^T-1h^t,p_γ(σ))^p
                +sup_γ∈ K∫_{x≥ R}x^p γ(dx)).

    

    
        Recall from Proposition <ref> that the relative compactness of K yields
        
    lim_σ↘ 0sup_γ∈ K∑_t=0^T-1h^t,p_γ(σ)=0,  lim_R→∞sup_γ∈ K∫_{x≥ R}x^p γ(dx)=0.

    

    
    Let μ, ν∈(F) for some bounded set F⊆ (^d)^T.
    
        
  *  There exists a constant C>0 that depends only on d, T, p, (F) such that for all σ>0,
        
    _p(μ, ν)^p
                ≤ C((1+σ)^p/σ_1(μ, ν)+(∑_t=0^T-1h^t,p_μ(σ)∨ h^t,p_ν(σ))^p).

        
  *  Further suppose that μ, ν∈_p, α, L((^d)^T) for some L>0 and 0<α≤ 1. Then there exists a constant C>0 that depends only on d, T, p, (F), L, α such that
        
    _p(μ, ν)
                ≤ C_1(μ, ν)^α^T-1/(pα^T-1+1).

    
    


    

 §.§ Comparison of topologies

    In this section, we complement the discussion in Section <ref> and Figure <ref> by giving concise mathematical statements.


     Let  μ∈_p((^d)^T) and (μ_n)_n∈⊆_p((^d)^T). Consider (σ_n)_n∈⊆ (0,∞) such that sup_n∈σ_n<∞.
    
        
  *  If ^(σ_n)_p(μ_n, μ)→ 0, then _p(μ_n, μ)→ 0. 
        
  *  If _p(μ_n, μ)→ 0 and _1(μ, μ_n)/σ_n→ 0, then ^(σ_n)_p(μ_n, μ)→ 0.
    
    

     Let  μ∈_p((^d)^T) and (μ_n)_n∈⊆_p((^d)^T). Consider (σ_n)_n∈⊆ (0,∞) such that sup_n∈σ_n<∞.
    
        
  *  If _p(μ_n, μ)→ 0, then ^(σ_n)_p(μ_n, μ)→ 0. 
        
  *  If ^(σ_n)_p(μ_n, μ)→ 0 and h^t,p_μ_n(σ_n)→ 0 for all t∈{1,2,…, T-1}, then _p(μ_n, μ)→ 0.
    
    

    
        Let σ>0. Then ^(σ)_p is a metric on _p((^d)^T) that induces the same topology as _p.
    


In conclusion, adding independent noise ξ fundamentally changes the temporal structure of stochastic processes and makes the smooth adapted Wasserstein topology different from the adapted Wasserstein topology in general. This contrasts with the Wasserstein topology, where the smooth Wasserstein distance generates the Wasserstein topology. In the case where the smoothing parameter is fixed for some σ>0, the smooth adapted Wasserstein distance completely ignores temporal evolution captured by the natural filtration.





§ PROOFS FOR SECTION <REF>


In this section we give proofs for the results of Section <ref>. Proofs for Section <ref> are contained in Section <ref> and proofs for Section <ref> can be found in Section <ref>.



 §.§ Proofs for Section <ref>


In this section, we first state and prove Lemma <ref>, which is essential for proving Proposition <ref>. We then present proofs of Proposition <ref>, Proposition <ref> and Proposition <ref>. We start by establishing some notation that will be used exclusively in the proofs of Lemma <ref> and Proposition <ref>.

Let μ∈_p((^d)^T). For t∈{1,2,…, T-1} and x_1:t∈ (^d)^t, we define μ̅_x_1:t∈_p((^d)^T-t) via

    μ̅_x_1:t(dx_t+1:T)
        =μ_x_1:T-1(dx_T)⋯μ_x_1:t+1(dx_t+2)μ_x_1:t(dx_t+1).

Equivalently, when X∼μ and A⊆ (^d)^T-t is measurable,

    μ̅_x_1:t(A)
        =(X_t+1:T∈ A  X_1:t=x_1:t).

Using this, we define ω̅_μ^t,p:(0,∞)→ [0,∞) via

    ω̅_μ^t,p(δ)
        =sup{([_p(μ̅_X_1:t, μ̅_Y_1:t)^p])^1/p |  X, Y∼μ, ([X_1:t-Y_1:t^p])^1/p<δ}.


        

    Let μ∈_p((^d)^T). For δ>0 and t∈{1,2,…, T-1}, we have
    
    ω̅_μ^t,p(δ)
            ≤∑_s=t^T-1 g_t,μ^s,p(δ)

    where
    
    g_t,μ^s,p(δ)
            :=ω_μ^s,p(δ+∑_ℓ=t^s-1 g_t,μ^ℓ, p(δ))  for  s∈{t,t+1 …, T-1}.

    Here, the sum ∑_ℓ=t^s-1 is understood as 0 when s=t.


    Let us fix t∈{1,2,…, T-1} and π∈(μ_1:t, μ_1:t) such that
    
    (∫x_1:t-y_1:t^p π(dx_1:t, dy_1:t))^1/p<δ.

    For s∈{t, t+1, …, T-1} and x_1:s, y_1:s∈ (^d)^s, define π^⋆_x_1:s, y_1:s∈(μ_x_1:s, μ_y_1:s) as a _p-optimal coupling between μ_x_1:s and μ_y_1:s. Note that γ∈(μ, μ) which is defined via
    
    γ(dx_1:T, dy_1:T)
       :=π^⋆_x_1:T-1, y_1:T-1(dx_T, dy_T)⋯π^⋆_x_1:t+1, y_1:t+1(dx_t+2, dy_t+2)π^⋆_x_1:t, y_1:t(dx_t+1, dy_t+1)π(dx_1:t, dy_1:t).

    Let (X, Y)∼γ. From the construction of γ, we bound
    
    ([X_s+1-Y_s+1^p])^1/p   =(∫x_s+1-y_s+1^p γ(dx_1:s+1, dy_1:s+1))^1/p
       =(∫_p(μ_x_1:s, μ_y_1:s)^p γ(dx_1:s, dy_1:s))^1/p
       ≤ω_μ^s, p(([X_1:s-Y_1:s^p])^1/p)
       ≤ω_μ^s,p(δ+∑_ℓ=t^s-1([X_ℓ+1-Y_ℓ+1^p])^1/p).

    Here the first inequality comes from the definition of ω_μ^s,p and the last inequality is due to the inequality (<ref>),
    
    ([X_1:t-Y_1:t^p])^1/p
            =(∫x_1:t-y_1:t^p π(dx_1:t, dy_1:t))^1/p
            <δ.

    Using the recursive formula (<ref>), we obtain that ([X_s+1-Y_s+1^p])^1/p≤ g_t,μ^s,p(δ). Observe that
    
    (∫_p(μ̅_x_1:t, μ̅_y_1:t)^p π(dx_1:t, dy_1:t))^1/p   ≤([X_t+1:T-Y_t+1:T^p])^1/p
       ≤∑_s=t^T-1([X_s+1-Y_s+1^p])^1/p≤∑_s=t^T-1g_t,μ^s,p(δ).

    Since the above estimate (<ref>) holds for all π∈(μ_1:t, μ_1:t) satisfying (<ref>), taking the supremum over all such π in the left hand side of (<ref>) gives the desired estimate.


    
    
        It follows from <cit.> that K⊆_p((^d)^T) is _p-relatively compact if and only if K is _p-relatively compact and
        
    lim_δ→ 0sup_μ∈ Kω̅_μ^t,p(δ)=0  for all  t∈{1,2,…, T-1}

        It is straightforward from the inequality _p(μ_x_1:t, μ_y_1:t) ≤_p(μ̅_x_1:t, μ̅_y_1:t) for all x_1:t, y_1:t∈ (^d)^t, that condition (<ref>) implies condition (<ref>). The fact that condition (<ref>) implies condition (<ref>) follows from Lemma <ref>.
    


    
    <ref> The proof is obvious from the definition of ω^t,p_μ.
    
    <ref>
        If k≤ 1, the proof is immediate from the fact that a map ω_μ^t,p(·) is nondecreasing. Now let us assume k>1. The proof for the case k>1 is a simple re-scaling of <cit.>. For the sake of simplicity, let us write =(^d)^t, =(^d) and define a map Ł_t:((^d)^T)→(×) via Ł_t(ν)=(Y_1:t, ν_Y_1:t) for Y∼ν. As a byproduct of <cit.>, we can relax the marginal constraints as follows:
        
    ω_μ^t,p(δ)
                =sup_γ∈(Ł_t(μ), δ)ρ^(γ)

        where
        
    ρ^(γ)=(∫x_1:t-y_1:t^p γ(dx_1:t, dκ, dy_1:t, dρ))^1/p,
       ρ^(γ)=(∫_p(κ, ρ)^p γ(dx_1:t, dκ, dy_1:t, dρ))^1/p,
       (Ł_t(μ), δ)
                ={γ∈^≤(×××) : ρ^(γ)<δ, P^(1)_#γ≤Ł_t(μ), P^(2)_#γ≤Ł_t(μ)}.

        Here, ^≤(×××) is the collection of positive measures whose total mass is no greater than 1 and the maps P^(j):×××→× are defined via P^(j)(x_1, y_1, x_2, y_2):=(x_j, y_j) for j∈{1,2}. Hence if γ∈(Ł_t(μ), kδ), then k^-pγ∈(Ł_t(μ), δ) and ρ^(γ)=kρ^(k^-pγ). This implies that ω_μ^t,p(kδ)≤ kω_μ^t,p(δ).
    

    
        Let μ∈_p, α, L((^d)^T). Suppose X, Y∼μ and [X_1:t-Y_1:t^p]<δ^p. Applying Jensen's inequality we obtain
        
    ([_p(μ_X_1:t, μ_Y_1:t)^p])^1/p≤ L([X_1:t-Y_1:t^pα])^1/p≤ L ([X_1:t-Y_1:t^p])^α/p≤ L δ^α.

        Hence, ω^t,p_μ(δ)≤ L δ^α.
    



 §.§ Proofs for Section <ref>

    
         The proof is a straightforward generalization of <cit.>: the case where the noise ξ is Gaussian is treated in <cit.> and <cit.> shows the same result when ξ is compactly supported.
         
         The fact that ^(σ)_p and ^(σ)_p satisfy symmetry and the triangle inequality is trivial; for example, see <cit.>. If ^(σ)_p(μ, ν)=0 or ^(σ)_p(μ, ν)=0, then μ∗ξ_σ=ν∗ξ_σ. By taking Fourier transforms, we deduce that
         
    μ(t)ξ(σ t)
                 =μ∗ξ_σ(t)
                 =ν∗ξ_σ(t)=ν(t)ξ(σ t)  for all  t∈ (^d)^T.

         Since the set of real zeros of ξ has Lebesgue measure zero, μ=ν a.s. In particular, μ=ν as tempered distributions. The Fourier inversion formula shows that μ=ν as tempered distributions and thus μ=ν as measures. This shows that ^(σ)_p and ^(σ)_p are metrics on _p((^d)^T). For a proof that ^(σ)_p metrizes the _p-topology we refer to <cit.>. Lastly, for σ_1, σ_2>0, find (X, Z)∼μ⊗ξ and (Y, W)∼ν⊗ξ such that (X+σ_1 Z, Y+σ_1 W) is an optimal coupling of ^(σ_1)_p(μ, ν). Then by the triangle inequality,
         
    ^(σ_2)_p(μ, ν)
                    ≤([(X+σ_2 Z)-(Y+σ_2 W)^p])^1/p
       ≤([(X+σ_1 Z)-(Y+σ_1 W)^p])^1/p
                 +σ_1-σ_2([Z-W^p])^1/p
       ≤^(σ_1)_p(μ, ν)+2σ_1-σ_2(M_p(ξ))^1/p.

         This proves desired estimates.
    








§ PROOFS FOR SECTION <REF>

Proofs for Sections <ref>, <ref>, <ref> and <ref> are contained in Sections <ref>, <ref>, <ref> and <ref>.



 §.§ Proofs for Section <ref>

This section is organized as follows. In Lemma <ref>, we show that compactly supported measures are _p-dense via an explicit estimate. Combined with Lemma <ref>, this proves Proposition <ref>. The proof of Theorem <ref> is based on Lemma <ref>. Following that, we prove Corollary <ref> and Corollary <ref>.


    Let R>0. There exists a Lipschitz function Φ^(R)=(Φ^(R)_1, …, Φ^(R)_T):(^d)^T→ (^d)^T such that
    
    _p(μ, Φ^(R)_#μ)
            ≤ 2T^2/p(∫_{x≥ R}x^p μ(dx))^1/p

    for any μ∈_p((^d)^T). Moreover, Φ^(R)_t≤ R for every t∈{1,2,…, T}.


    Throughout the proof, we assume that R>0 and μ∈_p((^d)^T) are fixed. We define Φ:(^d)^T→ (^d)^T via Φ(x_1:T):=(φ(x_1), …, φ(x_T)), where φ:^d→^d is given by
    
    φ(x)
            =
                x     if x< R
    Rx/x    if x≥ R.

    We want to construct a bicausal coupling between μ and Φ_#μ. Set π_1,1:=(𝕀, φ)_#μ_1∈(ν_1, (Φ_#ν)_1). For t∈{1,2,…, T-1} and x_1:t, y_1:t∈ (^d)^t, let us define π_x_1:t, y_1:t∈(^d×^d) via
    
    π_x_1:t, y_1:t
            :=
                (𝕀, φ)_#μ_x_1:t    if  x_1:t=y_1:t and x_ℓ<R  for all ℓ∈{1,2,…, t},
    μ_x_1:t⊗ (Φ_#μ)_y_1:t    otherwise.

    Since (Φ_#μ)_y_1:t=φ_#μ_x_1:t on {x_1:t=y_1:t and x_ℓ<R  for all ℓ∈{1,2,…, t}}, we have that π_x_1:t, y_1:t∈(μ_x_1:t, (Φ_#μ)_y_1:t). Define the measure π∈((^d)^T× (^d)^T) via 
    
    π(dx, dy)
            :=π_x_1:T-1, y_1:T-1(dx_T, dy_T)⋯π_x_1, y_1(dx_2, dy_2)π_1,1(dx_1, dy_1)

    and conclude that π∈(μ, Φ_#μ). Thus
    
    _p(μ, Φ_#μ)^p
            ≤∑_t=1^T∫x_t-y_t^p π(dx_1:t, dy_1:t).

    For t∈{1, 2, …, T}, consider the partition (E^t_ℓ)_ℓ=0^t of (^d)^t× (^d)^t where
    
    E^t_0:={(x_1:t, y_1:t): x_s<R  for all  s∈{1,2, …, t}},
       E^t_ℓ
            :={(x_1:t, y_1:t): x_ℓ≥ R  and x_s< R  for all  s∈{1,2, …, ℓ}} when ℓ∈{1,2,…, t}.

    Note that
    
    ∫_E^t_0x_t-y_t^p π(dx_1:t, dy_1:t)
            =∫_{x_t<R }x_t-φ(x_t)^p μ(x_1:t)=0.

    For ℓ∈{1,2, …, t}, we use Jensen's inequality to compute
    
    ∫_E^t_ℓx_t-y_t^p π(dx_1:t, dy_1:t)
               ≤ 2^p-1∫_E^t_ℓx_t^p μ(dx_1:t)
            +2^p-1∫_E^t_ℓy_t^p Φ_#μ(dy_1:t)
       ≤ 2^p∫_{x_ℓ≥ R}x_t^p μ(dx_1:t).

    Combining these results,
    
    _p(μ, Φ_#μ)^p
               ≤∑_t=1^T ∑_ℓ=1^t∫_E^t_ℓx_t-y_t^p π(dx_1:t, dy_1:t)
       ≤∑_t=1^T ∑_ℓ=1^t
            2^p∫_{x_ℓ≥ R}x_t^p μ(dx_1:t)
            ≤ 
            2^pT^2∫_{x≥ R}x^p μ(dx).



    
    For μ, ν∈((^d)^T) we have
    
    1/2μ-ν_≤(μ, ν)
            :=inf_π∈(μ, ν)π({x≠ y})
            ≤2^T-1/2μ-ν_.



    See <cit.>.


    
        Fix R>0 and define Φ^(R) as in Lemma <ref>. Observe that
        
    _p(Φ^(R)_#μ, Φ^(R)_#ν)^p
                   ≤(2R)^pinf_π∈(Φ^(R)_#μ, Φ^(R)_#ν)∫∑_t=1^T {x_t≠ y_t}π(dx_1:T, dy_1:T)
       ≤ (2R)^p T inf_π∈(Φ^(R)_#μ, Φ^(R)_#ν)∫{x_1:T≠ y_1:T}π(dx_1:T, dy_1:T)
       = (2R)^p T (Φ^(R)_#μ, Φ^(R)_#ν)
       ≤ (2R)^p T((2^T-1)/2) Φ^(R)_#μ-Φ^(R)_#ν_
       ≤ (2R)^p T((2^T-1)/2) μ-ν_.

        The first inequality stems from the inclusion {x_t≠ y_t}⊆{x_1:T≠ y_1:T}. Lemma <ref> gives the second inequality. The last inequality is straightforward from the definition of ·_. Indeed,
        
    1/2Φ^(R)_#μ-Φ^(R)_#ν_   =sup_A⊆ (^d)^Tμ((Φ^(R))^-1(A))-ν((Φ^(R))^-1(A))
       ≤sup_B⊆ (^d)^Tμ(B)-ν(B)
                =1/2μ-ν_.

        Using the triangle inequality together with the estimate (<ref>) and Lemma <ref>,
        
    _p(μ, ν)^p
                   ≤ 3^p-1(_p(μ, Φ^(R)_#μ)^p+_p(ν, Φ^(R)_#ν)^p+_p(Φ^(R)_#μ, Φ^(R)_#ν)^p)
       ≤ 3^p-12^p T^2 ∫_{x≥ R}x^p (μ+ν)(dx)
                +3^p-1(2R)^p T((2^T-1)/2) μ-ν_
       ≤ (3^p-12^p T^2)∨ (3^p-12^p T((2^T-1)/2)) (∫_{x≥ R}x^p (μ+ν)(dx)
                +R^pμ-ν_)
       ≤ 6^p-1T2^T(∫_{x≥ R}x^p (μ+ν)(dx)+R^pμ-ν_).

        This proves the desired estimate.
    
    

    
        For μ, ν∈_p((^d)^T) we have
        
    μ^σ-ν^σ_≤ f_L^1_1(μ, ν)/σ.

    
    
        We first claim that
        
    μ^σ-ν^σ_   =sup{∫φ dμ^σ-∫φ dν^σ : φ≤ 1}
       =sup{∫φ dμ^σ-∫φ dν^σ : φ≤ 1, ((φ))<∞}.

        The first equality is classical, see for example <cit.>. To prove the second equality, let us fix >0 and choose R>0 such that (μ^σ+ν^σ)({x∈ (^d)^T: x>R})≤. Take any function φ:(^d)^T→ such that φ≤ 1, set φ_R(x):=φ(x){x≤ R} and observe that
        
    ∫φ dμ^σ-∫φ dν^σ   =(∫φ_R dμ^σ-∫φ_R dν^σ)
                +(∫ [φ-φ_R] dμ^σ-∫ [φ-φ_R] dν^σ)
       ≤(∫φ_R dμ^σ-∫φ_R dν^σ)
                +.

        Since φ is arbitrary, it shows the claim.

        Now, let φ:(^d)^T→ such that φ≤ 1 and ((φ))<∞. Note that
        
    ∫φ dμ^σ-∫φ dν^σ
                =∫(∫φ(x)f_σ(x-y)dx)μ(dy)-∫(∫φ(x)f_σ(x-y)dx)ν(dy).

        Since φ has bounded support and f is continuously differentiable, the dominated convergence theorem implies that
        
    _y(∫φ(x)f_σ(x-y)dx)
                =-∫φ(x) ( f_σ)(x-y)dx.

        In particular
        
    _y(∫φ(x)f_σ(x-y)dx)≤∫ f_σ(x)dx
                =1/σ∫ f(x)dx.

        Thus, the function y↦∫φ(x)f_σ(x-y)dx is Lipschitz continuous with the Lispchitz constant f_L^1/σ. The desired estimate follows from the Kantorovich–Rubinstein duality for _1(μ, ν) and (<ref>).
    

    
    Combining Proposition <ref> and Lemma <ref>,
        
    _p(μ^σ, ν^σ)^p
                   ≤ 6^p-1T2^T (R^p μ^σ-ν^σ_
                +∫_{x≥ R}x^p (μ^σ+ν^σ)(dx))
       ≤ 6^p-1T2^T (R^p  f_L^1_1(μ, ν)/σ
                +∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)).

    

    
        <ref> For the sake of simplicity, let us write A_R:={x∈ (^d)^T : x≥ R}. Using the Markov inequality, we bound
        
    ∫_{x≥ R}x^p (μ^σ+ν^σ)(dx)
                   =∫x^p A_R(x)(μ^σ+ν^σ)(dx)
       ≤∫x^p x^q-p/R^q-p(μ^σ+ν^σ)(dx)
                =M_q(μ^σ+ν^σ)/R^q-p.

        Plugging this into Theorem <ref>,
        
    ^(σ)_p(μ, ν)^p
                ≤ 6^p-1T2^T (R^p f_L^1_1(μ, ν)/σ
                +M_q(μ^σ+ν^σ)/R^q-p).

        The desired estimate (<ref>) follows from optimizing the right-hand side by choosing
        
    R=(σ M_q(μ^σ+ν^σ)/ f_L^1_1(μ, ν))^1/q.


        <ref> Without loss of generality, we may assume that 0∈ ((μ)∪(ν))∩(ξ). The proof for the general case follows from translations. Applying the bound
        
    (M_q(μ^σ+ν^σ))^p/q   ≤ (M_q(μ^σ))^p/q
                +(M_q(ν^σ))^p/q
       ≤ ((M_q(μ))^1/q+σ (M_q(ξ))^1/q)^p
                +((M_q(ν))^1/q+σ (M_q(ξ))^1/q)^p
       ≤ 2(((μ)∪(ν))+σ((ξ)))^p

        and sending the limit q→∞ in (<ref>), we obtain
        
    ^(σ)_p(μ, ν)^p
                   ≤ 6^p-1T2^T+1lim inf_q→∞(M_q(μ^σ+ν^σ))^p/q( f_L^1_1(μ, ν)/σ)^1-p/q
       ≤ 6^p-1T2^T+2(((μ)∪(ν))+σ((ξ)))^p( f_L^1_1(μ, ν)/σ).


        <ref> If ξ is a Gaussian measure we have ξ_σ=(ξ_σ_0)∗ (ξ_√(σ^2-σ^2_0)). Since q>p, we apply Corollary <ref><ref> to deduce that
        
    ^(σ)_p(μ, ν)^p
                   =^(√(σ^2-σ^2_0))_p(μ^σ_0, ν^σ_0)^p
       ≤ 6^p-1T2^T+1(M_q(μ^σ+ν^σ))^p/q( f_L^1^(σ_0)_1(μ, ν)/√(σ^2-σ^2_0))^1-p/q.

    

    
        Choosing σ_0=σ/√(2) in Corollary <ref><ref>,
        
    ^(σ)_p(μ, μ_n)^p
                ≤ 6^p-1T2^T+1 f^1-p/q_L^1(M_q(μ^σ+μ_n^σ))^1/r(^(σ/√(2))_1(μ, μ_n)/σ/√(2))^1/r'

        where r=q/p>1 and r' is the Hölder conjugate of r. Applying Hölder's inequality with 1/r+1/r'=1, 
        
    [^(σ)_p(μ, μ_n)^p]
                   ≤ C(M_q(μ^σ)+[M_q(μ_n^σ)])^1/r([^(σ/√(2))_1(μ, μ_n)]/σ/√(2))^1/r'
       ≤ C(M_q(μ)+σ^q M_q(ξ))^p/q([^(σ/√(2))_1(μ, μ_n)]/σ)^1/r'
       ≤ C(1+σ^p)([^(σ/√(2))_1(μ, μ_n)]/σ)^1/r'

        for some constant C>0 that depends only on d, T, p, q, M_q(μ). A close inspection of the proof of <cit.> shows that if q>dT+2, then for X∼μ,
        
    [_1^(σ/√(2))(μ, μ_n)]
                   ≤∫_(^d)^Tx[f_σ/√(2)∗μ(x)-f_σ/√(2)∗μ_n(x)]dx
       ≤(∫_(^d)^Tx^2/1+x^q dx)^1/2(∫_(^d)^T(1+x^q)[f^2_σ/√(2)(x-X)]/n dx)^1/2
       ≤ C(1+σ^q/2/σ^dT/2)1/√(n).

        for some C>0 that depends only on d, T, q, M_q(μ). We refer to the proof of <cit.> for computational details. This proves the desired result.
    



 §.§ Proofs for Section <ref>


In Lemma <ref>, we show that there is a Monge mapping from μ^σ to μ whose kernels are almost _p-optimal. We use this to prove Theorem <ref>. We then present proofs of Corollaries <ref> and <ref>.

    
        Let μ∈_p((^d)^T) and σ>0. For each >0 there exists a bicausal coupling π^∈(μ^σ, μ) given by π^=(𝕀, G^)_#μ^σ, where G^=(G^_1, …, G^_T):(^d)^T→ (^d)^T is a Borel measurable function, and π^ satisfies
        
    (∫x_1-y_1^p π^(dx_1, dy_1))^1/p≤ (1+)_p((μ^σ)_1, μ_1)

        as well as
        
    (∫x_t+1-y_t+1^p π^_x_1:t,y_1:t(dx_t+1, dy_t+1))^1/p≤ (1+)_p((μ^σ)_x_1:t, μ_y_1:t)

        for all t∈{1,2, …, T-1} and x_1:t, y_1:t∈ (^d)^t. Moreover, G^_t depends only on the first t coordinates, i.e. G^_t(x_1:T)=G^_t(x_1:t) for t∈{1,2, …, T-1}.
    
    
        Throughout the proof, >0 is fixed. Since (μ^σ)_1 is absolutely continuous, it can be approximated by Monge mappings. Precisely speaking, from <cit.>,
        
    _p((μ^σ)_1, μ_1)^p
                =inf_T: T_#(μ^σ)_1=μ_1∫x_1-T(x_1)^p μ^σ(dx_1).

        Hence there exists a map F^_1:^d→^d such that (F^_1)_#(μ^σ)_1=μ_1 and
        
    ∫x_1-F^_1(x_1)^p μ^σ(dx_1)≤ (1+)^p_p((μ^σ)_1, μ_1)^p.

        Let us define π^_1,1:=(𝕀, F^_1)_#(μ^σ)_1∈((μ^σ)_1, μ_1). For t∈{1,2, …, T-1}, let us denote by π^⋆_x_1:t, y_1:t∈((μ^σ)_x_1:t, μ_y_1:t) a _p-optimal coupling for each x_1:t, y_1:t∈ (^d)^t. It can be chosen in such a way that
        
    (^d)^t× (^d)^t ∋ (x_1:t, y_1:t)
                ↦π^⋆_x_1:t, y_1:t∈(^d×^d)

        is Borel measurable, where (^d ×^d) is endowed with the weak topology, see e.g. <cit.>. Thanks to Lemma <ref>, the map (<ref>) can be approximated by parameterized Monge couplings. Applying Lemma <ref> with 𝒳=^d and 𝒵=(^d)^t× (^d)^t, we find a measurable map F^_t+1:(^d)^t× (^d)^t×^d→^d such that
        
    ∫x_t+1-F^_t+1(x_1:t, y_1:t, x_t+1)^p (μ^σ)_x_1:t(dx_t+1)
            ≤ (1+)^p_p((μ^σ)_x_1:t, μ_y_1:t )^p.

        Set π^_x_1:t, y_1:t:=(𝕀, F^_t+1(x_1:t, y_1:t, · ))_#(μ^σ)_x_1:t∈((μ^σ)_x_1:t, μ_y_1:t) and define π^∈(μ^σ, μ) via
        
    π^(dx_1:T, dy_1:T)
                =π^_x_1:T-1, y_1:T-1(dx_T, dy_T)⋯π^_x_1, y_1(dx_2, dy_2)π^_1,1(dx_1, dy_1).

        Note that if X∼μ^σ, then Y∼μ, where Y is defined via the following recursive relation,
        
    Y_1=F^_1(X_1)  and  Y_t+1=F^_t+1(X_1:t, Y_1:t, X_t+1)  for  t∈{1,2, …, T-1}.

        Substituting (<ref>) iteratively, the construction of G^ is now straightforward.
    

    
        Let >0 and choose π^=(𝕀, G^)_#μ^σ∈(μ^σ, μ) as in Lemma <ref>. Note that
        
    _p(μ^σ, μ)
                ≤(∫∑_t=1^T x_t-y_t^p π^(dx,dy))^1/p≤∑_t=1^T(∫x_t-y_t^p π^(dx, dy))^1/p.

        Using the fact that _p((μ^σ)_1, μ_1)≤ (M_p(ξ_σ))^1/p=σ (M_p(ξ))^1/p, we deduce from Lemma <ref> that
        
    I^_1
                :=(∫x_1-y_1^p π^(dx, dy))^1/p≤ (1+)_p((μ^σ)_1, μ_1)
                ≤ (1+)σ(M_p(ξ))^1/p.

        Now, fix t∈{1,2, …, T-1}. Since G^_t depends only on the first t coordinates, we denote (G_1^(x), …, G_t^(x)) by G^(x_1:t). Also, let us denote the density of (ξ_σ)_1:t by
        
    (f_σ)_1:t(x_1:t)
                :=∫_(^d)^T-t f_σ(x_1:T)dx_t+1:T.

        Using the fact that (ξ_σ)_z_1:t has the density
        
    z_t+1↦(f_σ)_1:t+1(z_1:t+1)/(f_σ)_1:t(z_1:t),

        we compute that for measurable B⊆^d,
        
    (μ^σ)_x_1:t(B)
                =∫ (f_σ)_1:t(x_1:t-z_1:t) (μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t)(B) μ(dz_1:t)/(f_σ)_1:t∗μ_1:t(x_1:t).

        A similar identity was used in  <cit.>. From Lemma <ref> and convexity of _p^p applied to (<ref>), we obtain
        
    ∫x_t+1-y_t+1^p π^(dx, dy)
       = ∫x_t+1-y_t+1^p π^_x_1:t, y_1:t (dx_t+1, dy_t+1) π^(dx_1:t, dy_1:t)
       ≤∫  (1+)^p _p((μ^σ)_x_1:t, μ_y_1:t)^p π^(dx_1:t, dy_1:t)
       = (1+)^p∫_p((μ^σ)_x_1:t, μ_G^(x_1:t))^p μ^σ(dx_1:t)
       ≤ (1+)^p ∫∫ (f_σ)_1:t(x_1:t-z_1:t)_p(μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t, μ_G^(x_1:t))^pμ (dz_1:t)/(f_σ)_1:t∗μ_1:t(x_1:t)μ^σ(dx_1:t).

        Since x_1:t↦ (f_σ)_1:t∗μ_1:t(x_1:t) is the density of (μ^σ)_1:t, we use the change of variable w_1:t=x_1:t-z_1:t to obtain
        
    ∫∫ (f_σ)_1:t(x_1:t-z_1:t)_p(μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t, μ_G^(x_1:t))^pμ(dz_1:t)/(f_σ)_1:t∗μ_1:t(x_1:t)μ^σ(dx_1:t)
       =∫∫ (f_σ)_1:t(x_1:t-z_1:t)_p(μ_z_1:t∗ (ξ_σ)_x_1:t-z_1:t, μ_G^(x_1:t))^p μ(dz_1:t) dx_1:t
       =∫∫_p(μ_z_1:t∗ (ξ_σ)_w_1:t, μ_G^(z_1:t+w_1:t))^p (μ_1:t⊗ (ξ_σ)_1:t)(dz_1:t, dw_1:t)
       =[_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_G^(X_1:t+σ Z_1:t))^p],

        where (X, Z)∼μ⊗ξ. Using the above, we bound
        
    I^_t+1:=(∫x_t+1-y_t+1^p π^(dx, dy))^1/p
       ≤ (1+) [_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_G^(X_1:t+σ Z_1:t))^p]^1/p
       ≤ (1+) ([_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_X_1:t)^p]^1/p+ [_p(μ_X_1:t, μ_G^(X_1:t+σ Z_1:t))^p]^1/p)
       ≤ (1+)(σ (M_p(ξ))^1/p+[_p(μ_X_1:t, μ_G^(X_1:t+σ Z_1:t))^p]^1/p)
       ≤ (1+)(σ (M_p(ξ))^1/p+ω_μ^t,p(X_1:t-G^(X_1:t+σ Z_1:t)_L^p))
       ≤ (1+)(σ (M_p(ξ))^1/p+ω_μ^t,p(σ (M_p(ξ))^1/p+X_1:t+σ Z_1:t-G^(X_1:t+σ Z_1:t)_L^p))
       ≤ (1+)(σ (M_p(ξ))^1/p+ω_μ^t,p(σ (M_p(ξ))^1/p+∑_s=1^tI^_s)).

        Here, the third inequality is immediate from
        
    [_p(μ_X_1:t∗ (ξ_σ)_σ Z_1:t, μ_X_1:t)^p]
                   ≤[∫z_t+1^p (ξ_σ)_σ Z_1:t(dz_t+1)]
       =[∫z_t+1^p (f_σ)_1:t+1(σ Z_1:t, z_t+1)/(f_σ)_1:t(σ Z_1:t)dz_t+1]
       =∫∫z_t+1^p (f_σ)_1:t+1(z_1:t+1)/(f_σ)_1:t(z_1:t)dz_t+1(f_σ)_1:t(z_1:t)dz_1:t
       =σ^p M_p(ξ_t+1).

        
        
        
        
        
        The definition of ω_μ^t,p justifies the fourth inequality. The fifth inequality is the triangle inequality and monotonicity of ω_μ^t,p. To see the last inequality, note that (X+σ Z, G^(X+σ Z))∼π^ and
        
    X_1:t+σ Z_1:t-G^(X_1:t+σ Z_1:t)_L^p
                =(∫x_1:t-y_1:t^p π^(dx_1:t, dy_1:t))^1/p≤∑_s=1^t I^_s.

        Starting from the initial estimate (<ref>), we bound I^_t by h^t-1, p_μ using the recursive estimate (<ref>) along with Proposition <ref><ref> as follows,
        
    lim inf_→ 0I^_t≤ (1∨ (M_p(ξ))^1/p)(h^0,p_μ(σ)+th^t-1,p_μ(σ))

        for all t∈{2, 3,…, T}. This can be shown by the induction. When t=2, we compute from (<ref>) and (<ref>) that 
        
    lim inf_→ 0I^_2
                   ≤σ (M_p(ξ))^1/p+ω_μ^1,p(σ (M_p(ξ))^1/p+I^_1)
       ≤σ (M_p(ξ))^1/p+ω_μ^1,p(2σ (M_p(ξ))^1/p)
                ≤ (1∨ (M_p(ξ))^1/p)(σ+2ω^1,p_μ(σ)).

        Here, the last inequality follows from Proposition <ref><ref>. This proves (<ref>) when t=2. Assuming (<ref>) holds for I^_s, s∈{2,…, t}, a similar computation shows that
        
    lim inf_→ 0 I^_t+1   ≤σ(M_p(ξ))^1/p+ω^t,p_μ(σ(M_p(ξ))^1/p+I^_1+∑_s=2^tI^_s)
       ≤σ(M_p(ξ))^1/p
                +ω^t,p_μ(2σ(M_p(ξ))^1/p+∑_s=2^t(1∨ (M_p(ξ))^1/p)(h^0,p_μ(σ)+sh^s-1,p_μ(σ)))
       ≤σ(M_p(ξ))^1/p
                +(t+1)(1∨ (M_p(ξ))^1/p)ω^t,p_μ(∑_s=0^t-1h^s,p_μ(σ)).

        Hence (<ref>) holds for all t∈{2,3,…, T}.
        Plugging this back into (<ref>),
        
    _p(μ^σ, μ)
                ≤lim inf_→ 0∑_t=1^TI^_t≤σ(M_p(ξ))^1/p+∑_t=2^T(1∨ (M_p(ξ))^1/p)(h^0,p_μ(σ)+th^t-1,p_μ(σ)).

        This proves the desired estimate.
    


    
        <ref>  From Proposition <ref>, ω^t,p_μ(σ)≤ L σ^α if μ∈_p, α, L((^d)^T). Note from the construction of h^t,p_μ that h^1,p_μ(σ)=ω^1,p_μ(σ)≤ Lσ^α and
        
    h^2,p_μ(σ)
                =ω^2,p_μ(σ+h^1,p_μ(σ))
                ≤ L(σ+Lσ^α)^α≤ L(σ^α+L^ασ^α^2)
                ≤ (L∨ L^α+1)(σ^α+σ^α^2).

        It is straightforward from the mathematical induction that h^t,p_μ(σ)≤ C (σ^α+σ^α^2+…+σ^α^t) for some C>0 that depends only on L, α if t∈{1,2,…, T-1}. Applying this bound to Theorem <ref> yields the desired estimate.

        <ref>  From Theorem <ref>, we find
        
    sup_μ∈ K_p(μ^σ, μ)
                ≤ T(1∨ (M_p(ξ))^1/p)∑_t=0^T-1h^t,p_K(σ)

        where we define h^0,p_K(σ):=σ and h^t,p_K(σ):=sup_μ∈ Kω^t,p_μ(∑_s=0^t-1 h^s,p_K(σ)), t∈{1,2,…, T-1}. Note that if sup_μ∈ Kω^t,p_μ(δ)→ 0 as δ→ 0, the right hand side of (<ref>) goes to 0 as σ→ 0. The last statement is a straightforward application of a triangle inequality.
    

    
        From Corollaries <ref> and <ref><ref>, we estimate
        
    [_p(μ, (μ_n)^σ)^p]
                   ≤ C([^(σ)_p(μ, μ_n)^p]+[_p(μ^σ, μ)^p])
       ≤ C(1+σ^(q+p)/2/σ^(1+dT/2)(1-p/q))n^-(q-p)/(2q)
                +C∑_t=0^T-1σ^pα^t.

        We optimize the convergence rate of the right hand side by choosing σ = n^-γ where
        
    γ=1-p/q/(dT+2)(1-p/q)+2pα^T-1.

    




 §.§ Proofs for Section <ref>

      
          This is straightforward from the inequality
          
    _p(μ, ν)^p
                  ≤ 3^p-1^(σ)_p(μ, ν)
                  +3^p-1_p(μ, μ^σ)
                  +3^p-1_p(ν, ν^σ)

          applied to Theorem <ref> and Theorem <ref>.
      

      
          Choose ξ to be supported on {x∈ (^d)^T : x≤ 1}. From 0<σ<R, we have
          
    ∫_{x≥ 2R}x^p μ^σ(dx)
                     =∫_{x+σ z≥ 2R}x+σ z^p μ(dx)ξ(dz)
       ≤∫_{x≥ R}x+σ z^p μ(dx)ξ(dz)
                  ≤ 2^p-1∫_{x≥ R}x^p μ(dx)+2^p-1σ^p.

          Applying this bound to Proposition <ref> yields the desired result.
      
      
      
      <ref> By a translation argument we may assume that 0∈ F. Let μ, ν∈(F). Choose ξ to be supported on {x∈ (^d)^T : x≤ 1}. By combining the triangle inequality with Corollary <ref><ref> and Theorem <ref>, we compute
      
    _p(μ, ν)
                 ≤^(σ)_p(μ, ν)
              +2(_p(μ, μ^σ)∨_p(ν, ν^σ))
       ≤ C((1+σ)(_1(μ, ν)/σ)^1/p
              +∑_t=0^T-1h^t,p_μ(σ)∨ h^t,p_ν(σ))

      for some C>0 that depends only on d, T, (F). This proves the desired estimate.

      <ref> Let μ, ν∈(F)∩_p, α, L((^d)^T). Using Corollary <ref><ref> and Corollary <ref><ref>, we estimate
        
    _p(μ, ν)
                ≤ C( (1+σ)(_1(μ,ν)/σ)^1/p
                +∑_t=0^T-1σ^α^t)

        with some constant C>0 that depends only on d, T, L, α, (F). In (<ref>), choose σ=_1(μ, ν)^β for β=(pα^T-1+1)^-1 and we bound the right-hand side by
        
    (1+σ)(_1(μ,ν)/σ)^1/p
                +∑_t=0^T-1σ^α^t
       =(1+_1(μ, ν)^β)_1(μ,ν)^α^T-1/(pα^T-1+1)
                +∑_t=0^T-1_1(μ,ν)^α^t/(pα^T-1+1)
       ≤(1+(2(F))^β+∑_t=0^T-1(2(F))^(α^t-α^T-1)/(pα^T-1+1))_1(μ,ν)^α^T-1/(pα^T-1+1).

        Here, we make use of the fact that _1(μ, ν)≤ M_1(μ)+M_1(ν)≤ 2(F). This proves the desired result.
    



 §.§ Proofs for Section <ref>

In this section we prove Theorem <ref>, Theorem <ref> and Theorem <ref>. All the proofs are based on the following simple reformulation of the sequential convergence.


    Let  be a metric space. Suppose (x_n)_n∈⊆ and x∈. Then x_n→ x in  if and only if for every subsequence (x_n_k)_k∈ of (x_n)_n∈, there exists a subsequence of (x_n_k)_k∈ that converges to x.



    <ref> Suppose ^(σ_n)_p(μ_n, μ)→ 0. From Lemma <ref>, by passing to a subsequence, it is sufficient to assume that σ_n→σ_0 for some σ_0∈ [0,∞). From Proposition <ref>,
    
    ^(σ_0)_p(μ_n, μ)
            ≤_p^(σ_n)(μ_n, μ)
            +2σ_n-σ_0(M_p(ξ))^1/p≤_p^(σ_n)(μ_n, μ)
            +2σ_n-σ_0(M_p(ξ))^1/p.

    Since the right hand side converges to 0 as n→∞, we conclude that μ_n→μ in ^(σ_0)_p. We conclude that μ_n→μ in _p thanks to Proposition <ref>.


    <ref> Suppose now that _p(μ_n, μ)→ 0 and _1(μ_n, μ)/σ_n→ 0. From Lemma <ref>, we may assume that σ_n→σ_0 for some σ_0∈ [0,∞). We find from Theorem <ref> that for any R>0,
        
    lim sup_n→∞^(σ_n)_p(μ_n, μ)
                ≤ 6^p-1T2^T sup_n∈∫_{x≥ R}x^p (μ_n^σ_n+μ^σ_n)(dx).

        Note that μ^σ_n_n→μ^σ_0 and μ^σ_n→μ^σ_0 in _p. In particular, {μ^σ_n_n, μ^σ_n : n∈} is _p-relatively compact. Thus, the right hand side of (<ref>) can be made arbitrarily small as R→∞. This shows that ^(σ_n)_p(μ_n, μ)→ 0.



    <ref>  Suppose _p(μ_n, μ)→ 0. As in the proof of Theorem <ref>, it is sufficient to prove ^(σ_n)_p(μ_n, μ)→ when σ_n→σ_0 for some σ_0∈ [0,∞). If σ_0>0, then _1(μ_n, μ)/σ_n→ 0. By Theorem <ref><ref> this implies that ^(σ_n)_p(μ_n, μ)→ 0. Suppose now that σ_0=0. By the triangle inequality,
    
    ^(σ_n)_p(μ_n, μ)
            ≤_p(μ_n, μ)
            +2sup_ν∈ K_p(ν^σ_n, ν)

    where K={μ, μ_n :  n∈}. Note that K is _p-relatively compact. Thus, Proposition <ref> and Corollary <ref><ref> show that sup_ν∈ K_p(ν^σ_n, ν)→ 0. In conclusion, ^(σ_n)_p(μ_n, μ)→ 0.

    <ref> Suppose ^(σ_n)_p(μ_n, μ)→ 0 and h^t,p_μ_n(σ_n)→ 0 for all t∈{1,2,…, T-1}. Since ω^t,p_μ_n≤ h^t,p_μ_n, we have ω^t,p_μ_n(σ_n)→ 0. Using Lemma <ref>, we may assume that σ_n→σ_0 for some σ_0∈ [0,∞). If σ_0>0, fix >0 and choose N∈ such that sup_n≥ Nω^t,p_μ_n(σ_n)≤ and inf_n≥ Nσ_n≥σ_0/2. Note that
    
    lim sup_δ→ 0sup_n∈ω^t,p_μ_n(δ)
            ≤lim sup_δ→ 0∑_n=1^Nω^t,p_μ_n(δ)
            +sup_n≥ Nω^t,p_μ_n(σ_0/2)
            ≤ 0+sup_n≥ Nω^t,p_μ_n(σ_n)
            ≤.

    Since >0 was arbitrary and _p(μ_n, μ)→ 0, we deduce from Proposition <ref> that {μ_n :  n∈} is _p-relatively compact. Thus, _p(μ_n, μ)→ 0.

    Now, suppose σ_0=0. Then by the triangle inequality,
    
    lim sup_n→∞_p(μ_n, μ)
               ≤lim sup_n→∞(^(σ_n)_p(μ_n, μ)
            +_p(μ_n^σ_n, μ_n)
            +_p(μ^σ_n, μ))
       =lim sup_n→∞_p(μ_n^σ_n, μ_n).

    Here, the equality follows from Corollary <ref><ref> applied to K={μ}. It is easy to check from Theorem <ref> that h^t,p_μ_n(σ_n)→ 0 for all t∈{1,2,…, T-1} implies that _p(μ^σ_n_n, μ_n)→ 0. Hence, we conclude that _p(μ_n, μ)→ 0.






    From Proposition <ref> and the fact that ^(σ)_p≤ C^(σ)_p, it is obvious that if μ_n→μ in ^(σ)_p, then μ_n→μ in _p. Conversely, let us assume _p(μ_n, μ)→ 0. By choosing σ_n≡σ in Theorem <ref><ref>, we obtain that ^(σ)_p(μ_n, μ)→ 0. Therefore, ^(σ)_p metrizes the _p-topology. 














§ PROOF REGARDING EXAMPLE <REF>

    
        Let (_n)_n∈, (σ_n)_n∈⊆ (0, ∞) such that _n→ 0 and sup_n∈σ_n<∞. We define μ_n:=1/2δ_(_n, 1)+1/2δ_(-_n, -1), μ:=1/2δ_(0,1)+1/2δ_(0,-1), and ξ_σ_n:=(0, σ^2_n_2). Then we have
        
            
  *  _n/σ_n→ 0 if and only if _1(μ_n, μ)/σ_n→ 0 if and only if ^(σ_n)_p(μ_n, μ)→ 0.
            
  *  _n/σ_n→∞ if and only if ω^1,p_μ_n(σ_n)→ 0 if and only if _p(μ_n^σ_n, μ_n)→ 0.
        
    

    
        <ref> It is evident from _p(μ_n, μ)=_n that _n/σ_n→ 0 if and only if _1(μ_n, μ)/σ_n→ 0. The fact that _1(μ_n, μ)/σ_n→ 0 implies ^(σ_n)_p(μ_n, μ)→ 0 follows from Theorem <ref><ref>. Let us assume that ^(σ_n)_p(μ_n, μ)→ 0. We compute (μ^σ_n)_1=N(0, σ_n^2), (μ^σ_n)_x=1/2(1, σ_n^2)+1/2(-1, σ_n^2) and
        
    (μ^σ_n_n)_1=1/2(_n, σ_n^2)+1/2(-_n, σ_n^2),
       (μ^σ_n_n)_y
            =φ_σ_n(y-_n)/φ_σ_n(y+_n)+φ_σ_n(y-_n)(1,σ_n^2)+φ_σ_n(y+_n)/φ_σ_n(y+_n)+φ_σ_n(y-_n)(-1, σ^2_n)

        where φ(x):=(2π)^-1/2e^-x^2/2 and φ_σ_n(·):=(σ_n)^-1φ(·/σ_n). Since (μ^σ_n)_x does not depend on x, 
        
    ^(σ_n)_p(μ, μ_n)^p
               =inf_γ∈((μ^σ_n)_1, (μ^σ_n_n)_1)∫_^2x-y^p +_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p γ(dx, dy)
       =_p((μ^σ_n)_1, (μ^σ_n_n)_1)^p
            +∫_-∞^∞_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p(μ^σ_n_n)_1(dy).

    Using a change of variable, we write
    
    ∫_-∞^∞_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p(μ^σ_n_n)_1(dy)
       =∫_-∞^∞_p((1, σ^2_n)/2+(-1,σ^2_n)/2, a_n(y)(1, σ^2_n)+b_n(y)(-1,σ^2_n))^pφ(y)dy,

    where a_n(y):=φ(y)/(φ(y)+φ(y+2_n/σ_n)) and b_n(y):=1-a_n(y). If _n/σ_n↛ 0, then up to a subsequence, we may assume that _n/σ_n→ c for some c∈ (0, ∞]. From (<ref>) and the Fatou lemma,
    
    0=lim_n→∞_p^(σ_n)(μ, μ_n)^p
               ≥lim inf_n→∞∫_-∞^∞_p((μ^σ_n)_x, (μ_n^σ_n)_y)^p(μ^σ_n_n)_1(dy)
       = lim inf_n→∞∫_-∞^∞1/2 (_p((μ^σ_n)_x, (μ_n^σ_n)_ϵ_n+σ_n y)^p + _p((μ^σ_n)_x, (μ_n^σ_n)_-ϵ_n+σ_n y )^p ) φ(y)dy
       ≥∫_-∞^∞_p(δ_1+δ_-1/2, φ(y)δ_1+φ(y+2c)δ_-1/φ(y)+φ(y+2c))^pφ(y)dy
            >0.

    Thus, ^(σ_n)_p(μ, μ_n)→ 0 only if _n/σ_n→ 0. 
    
    <ref> It is easy check that ω^1,p_μ_n(σ_n)=(σ_n/_n)∧ 2. Hence _n/σ_n→∞ if and only if ω^1,p_μ_n(σ_n)→ 0. From Theorem <ref>, we have _p(μ^σ_n_n, μ_n)
        ≤ C (σ_n+ω^1,p_μ_n(σ_n)). Hence ω^1,p_μ_n(σ_n)→ 0 yields _p(μ^σ_n_n, μ_n)→ 0. Now, let us assume that _p(μ^σ_n_n, μ_n)→ 0. Note that
    
    _p(μ_n^σ_n, μ_n)^p
            =inf_γ∈((μ_n^σ_n)_1, (μ_n)_1)∫_^2x-y^p +_p((μ_n^σ_n)_x, (μ_n)_y)^pγ(dx, dy).

    Choose γ^⋆ to be optimal for (<ref>). Let Z∼(0,1), c_n(x):=φ_σ_n(x-_n)/(φ_σ_n(x-_n)+φ_σ_n(x+_n)) and d_n(x):=1-c_n(x). Then
    
    _p(μ_n^σ_n, μ_n)^p
               ≥∫_{y=_n}_p((μ_n^σ_n)_x, δ_1)^pγ^⋆(dx, dy)
            +∫_{y=-_n}_p((μ_n^σ_n)_x, δ_-1)^pγ^⋆(dx, dy)
       =∫_{y=_n} c_n(x)[σ_n Z^p]+d_n(x)[2+σ_n Z^p] γ^⋆(dx, dy)
       +∫_{y=-_n} c_n(x)[2+σ_n Z^p]+d_n(x)[σ_n Z^p] γ^⋆(dx, dy)
       ≥ ([σ_n Z^p]+[2+σ_n Z^p])∫_-∞^∞ (c_n(x)∧ d_n(x))(μ^σ_n_n)_1(dx).

    We compute
    
    ∫_-∞^∞ (c_n(x)∧ d_n(x))(μ^σ_n_n)_1(dx)
               =1/2∫_-∞^∞ (φ_σ_n(x-_n)∧φ_σ_n(x+_n))dx
       =1/2(∫_-∞^-_n/σ_n+∫__n/σ_n^∞)φ(x)dx.

    Hence _p(μ^σ_n_n, μ_n)→ 0 implies σ_n/_n→∞.
    


    



§ MONGE MAPS


    Let 𝒜, ℬ, 𝒵 be Polish spaces. We say that π is a kernel from 𝒵 to 𝒜 if π is a Borel measurable map from 𝒵 to (𝒜). Here, (𝒜) is endowed with the weak topology. We will denote the probability measure π(z) by π^z. We use a similar notation for functions. Given a Borel measurable function T:𝒵×𝒜→ℬ, we define T^z:𝒜→ℬ via T^z(a)=T(z,a). For a partition ℛ of 𝒳, we define its mesh via ℛ=sup_A∈ℛ(A).

    
        Let 𝒳,𝒵 be Polish spaces and let π be a kernel from 𝒵 to 𝒳×𝒳. Let μ^z and ν^z denote the pushforward measure of π^z through a projection on to the first coordinate and the second coordinate, respectively, i.e. π^z∈(μ^z, ν^z). Suppose μ^z, ν^z∈_p(𝒳) and μ^z has no atoms for all z∈𝒵. Then there exist Borel measurable maps T_n:𝒵×𝒳→𝒳 such that (𝕀, T^z_n)_#μ^z∈(μ^z, ν^z) and
        
    ∫_𝒳×𝒳 d_𝒳(x,y)^p π^z(dx, dy)
                =lim_n→∞∫_𝒳 d_𝒳(x, T_n^z(x))^p μ^z(dx).

        Here d_𝒳 is a metric on 𝒳 that induces the Polish topology.
    
    
        It follows from <cit.> that there exits a Borel measurable map T:𝒵×𝒳× [0,1]→𝒳× [0,1] such that
        
            
  * T^z:=T(z, ·):𝒳× [0,1]→𝒳× [0,1] is a Borel isomorphism, i.e. T^z is Borel measurable, bijective and its inverse is also Borel measurable,
            
  * for the Lebesgue measure λ on [0,1], (𝕀, T^z)_#(μ^z⊗λ)∈(μ^z⊗λ, ν^z⊗λ) and its projection on 𝒳×𝒳 is π^z.
        
        Now let us choose a partition ℛ_n of 𝒳 that consists of at most countable sets and satisfies lim_n→∞ℛ_n=0. We apply <cit.> to find Borel measurable maps Φ_n:𝒵×𝒳→𝒳× [0,1] such that
        
        
            
  * Φ_n^z:=Φ_n(z, ·):𝒳→𝒳× [0,1] is Borel isomorphism,
            
  * (Φ^z_n)_#(μ^z|_A)=(μ^z|_A)⊗λ for all A∈ℛ_n where μ^z|_A(·):=μ^z( ·∩ A).
        
        Let P_𝒳 and P_𝒵 be projection maps on 𝒳 and 𝒵 respectively and define a map T_n:𝒵×𝒳→𝒳 via T_n = P_𝒳∘ T ∘ (P_𝒵, Φ_n). Note that π^z_n:=(𝕀, T^z_n)_#μ^z∈(μ^z, ν^z) by (b) and (d). Similarly as in the proof of <cit.>, we can show that
        
    π^z_n(A× B)
                =π^z(A× B)  for any  A, B∈ℛ_n  and  z∈𝒵.

        Indeed, for A, B∈ℛ_n,
        
    π^z_n(A× B)
                =μ^z(A∩ (T^z_n)^-1(B))
                   =μ^z|_A((Φ^z_n)^-1∘ (T^z)^-1∘ (P_𝒳)^-1(B))
       =μ^z|_A⊗λ ((T^z)^-1∘ (P_𝒳)^-1(B))
       =μ^z⊗λ ((A× [0,1])∩ (T^z)^-1(B× [0,1]))
       =(𝕀, T^z)_#(μ^z⊗λ)((A× [0,1])× (B×[0,1]))
       =π^z(A× B).

        The first equality comes from the definition of π_n^z. For the second equality, we use the definition of T^z_n. The property (d) implies the third equality. The fourth and the fifth equality are straightforward from the pushforward operator and the last equality is obtained from the property (b). Thus, we deduce from <cit.> that π^z_n →π^z in _p on (𝒳×𝒳)^2. This implies the desired results.
    





abbrv





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



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


Open Universiteit, the Netherlands

University of Cape Town, South-Africa

    August 5, 2024
================================================================================================



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






§ INTRODUCTION


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


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


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






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





 §.§ Preliminaries


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

    ∫_ℤ_pdx=1

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


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

    V(𝒢)→ℚ_p,

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

    B_r(I)={x∈ℚ_p: |x-I|≤ r}

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

    K_N =_I∈ V(𝒢)B_p^-N(I).

Each vertex of 𝒢 can be assumed to have the form 

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

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




 §.§ Basis of L^2(K_N,ℂ)


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

    ⟨ f, g ⟩ =∫_K_N f(x)  g(x)   dx

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

    Ω(p^N|x-I|_p)=:Ω_I,N(x)

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


The functions 

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

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

Proof. We have that p^-r≤ p^-N, and 

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

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

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

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

Therefore an orthonormal basis of 

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

is the set of functions of the form 

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




We have L^2(K_N,ℂ)=X_N ⊕ℒ_0^2(K_N).


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






    ⟨Ω_I,N,Ψ_j,J,r⟩=0,

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

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




 §.§ Two operators over L^2(K_N,ℂ)

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

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

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

    𝒜 u(x)=∫_K_N A(x,y)u(y)dy.

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

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

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

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

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

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

Recall that the Laplacian matrix L is defined as 

    L=A-(γ_Iδ_IJ)

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

    ℒu(x)=∫_K_NL(x,y)u(y) dy

with

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

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


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

    𝕃u(x)=∫_K_NA(x,y) (u(y)-u(x))dy

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

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

where

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

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

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

On the other hand, we have 

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

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

    ℒf(x_0)=-p^N-rγ_J_0,

and

    𝕃f(x_0)=0.

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

    ℒu(x)=0,

and

    𝕃u(x)=𝒜u(x)-u(x)γ(x)=-u(x)γ(x).

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



The operator ℒ is compact.


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


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

    𝕃u(x)=𝒜u(x)-u(x)γ(x),

where 

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

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


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








𝕃 is not compact.



This is an immediate consequence of Proposition <ref>.


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



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





 §.§ Non-autonomous p-adic Zúñiga operators

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

    𝕃(t)=𝒜(t)-Γ(t)

where

    𝒜(t)u(x)=∫_K_NA(x,y,t)u(y)dy,

and 

    Γ(t)u(x)=u(x)γ(t)(x),

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

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

and

    γ(t)(x)=∑_I∈ V(𝒢)γ_I(t)Ω_I,N(x)

with 
γ_I(t)=∑_J∈ V(𝒢)A_IJ(t).


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



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

Proof. Let u∈ L^2(K_N,ℂ)  then

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

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

    |t-s|<δ|A_IJ(t)-A_IJ(s)|<ε/n^2.

Hence

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

Therefore

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

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

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

Hence, 

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

Finally, we have 

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

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


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

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

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

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

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




 §.§ Dirichlet and von Neumann eigenvalues


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

    Ω_E(t)=(x,y)∈ K_N^2| A(x,y,t)≠ 0,

as well as the p-adic vertex boundary:

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

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

    ∂_NS(t)=(x,y)∈Ω_E(t)| x∈ S∧ y∉ S

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

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

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

    V(S):=I∈ V| B_p^-N(I)∩ S≠∅.

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

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

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

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

again with fixed t≥0.


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

    D_∂ S(t),+f(x)=0

where 

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

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


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

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

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

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

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


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

    λ_1,𝒢^(D)(t)  and  λ_S,𝒢^(N)(t),

respectively.



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



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

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

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

    ∫_K_Nψ(x) dx=0,

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



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


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

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

where

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

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



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

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

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


The following examples show that the inequalities can be strict:


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

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

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

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

for given t≥0.



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

    ∫_Sf(x)^2γ(x)(t) dx=0

This means that 

    λ_S,𝒢(t)^(N)(t)=∞

and consequently

    λ_S,𝕃^(N)(t)<λ_S,𝒢^(N)(t)

because γ̂_S(t)<∞.








§ NON AUTONOMOUS CAUCHY PROBLEM


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




 §.§ Review of the abstract Cauchy problem





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


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

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


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

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

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

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


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

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




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



<cit.>.


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

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

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


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

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

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

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

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

    U(t,s)=e^∫_s^tA(τ) dτ

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



<cit.>.





 §.§ Reviewing  the autonomous case for graphs



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


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

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


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

    (L) {-γ_I: I∈ V(𝒢)}

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

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

where the functions φ_μ(x) are defined by: 

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

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

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





<cit.>.


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

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

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

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

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


This proves that 

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

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

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

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





 §.§ Approximating the non-autonomous case for graphs


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

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

We have the following proposition:


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

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

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



<cit.>.



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

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

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


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

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

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

Now it is clear that 

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

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


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


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

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

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

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

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

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

Note that  for each s≥ 0, the set 

    φ_μ(x,s)=φ_μ(s)(x)|μ(s)∈(L(s))

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

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

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

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

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

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

Thus,

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

and 

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

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

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

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

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

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

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

and 

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

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

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

and 

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

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

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

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

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

For simplicity, we rename the approximation above: 

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

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

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

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

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

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

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

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

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

Hence, by Parseval's equality we obtain:

    lim_n→∞ Q_μ(n,t,s)=Q_μ(t,s)

and 

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

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

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


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

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


This proves the following:



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

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

where t≥ s,

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

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

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

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



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

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

for t≥ s≥0.



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



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



































§ INHOMOGENEOUS MARKOV PROCESSES AND FELLER EVOLUTION.





 §.§ Reviewing inhomogeneous Markov processes

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


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

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

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

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

    ℱ_t^τ=σ(X_s:τ≤ s ≤ t)

with 0≤τ≤ t ≤ T. 


We recall some definitions:



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

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


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



The process

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

is called a time-inhomogeneous Markov process if 

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

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


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

    ℱ_t^τ=σ(X_s:τ≤ s ≤ t)

with 0≤τ≤ t ≤ T and 

    ℙ_τ,x(X_t∈ A)=P(τ,x;t,A)

with 0≤τ≤ t ≤ T, A∈ℰ.






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

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


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

    P(τ,t)f(x)=∫ P(τ,x;t,dy)f(y)

for f∈ C_b(E). Where the mapping 

    (τ,x,t,B)↦ P(τ,x;t,B)

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




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



<cit.>.





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


In this section we prove that the system 

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

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

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

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


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


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

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

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

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

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


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

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


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



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



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

    u(x,t)=∫_K_N u_s(x) P(t,x;s,dy).

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

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

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

Let ε>0, and δ>0 such that 

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

We have that

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

and 

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

If ||u||_∞≤ 1, then

    ||𝕃(t)u-𝕃(s)u||_∞≤ε((K_N)+1).

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




§ ACKNOWLEDGEMENTS


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

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



plain










 Paper Title: Conditional Independence in Stationary Diffusions.



















§ ABSTRACT 
Stationary distributions of multivariate diffusion processes have
recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The  result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift's sparsity pattern.



Constraints on large-scale polarization in northern hemisphere
    Chang Feng
    August 5, 2024
==============================================================













§ INTRODUCTION
 




Consider an ℝ^-valued stationary stochastic process {𝕏(t)}_t≥ 0 that arises as the solution 
of a stochastic differential equation (SDE)

    d𝕏(t)= f(𝕏(t)) dt + D  d𝕎(t),

 understood in the Itô sense <cit.>.
Here,  f: ℝ^→ℝ^ is a Lipschitz-continuous map that specifies the drift, D∈ℝ^× is a diagonal and invertible matrix, and 𝕎 is a standard -dimensional Brownian motion.  Assuming its unique existence, the stationary (marginal) distribution of the process 𝕏 depends on D only through the diffusion matrix C =  DD^T, and we denote the distribution by P_f,C.  Our interest lies in examining the conditional independence (CI) relations that hold in P_f,C when the drift is sparse, i.e., when each component function f_i only depends on some of its input coordinates.

Let X be an ℝ^-valued random vector  distributed according to P_f,C.  Let I,J,K⊆[]:={1,…,} be disjoint index sets, and denote subvectors of X as X_I=(X_i)_i∈ I.  Then we ask which sparsity restrictions on f imply that X_I and X_J are conditionally independent given X_K; in symbols, X_I X_J| X_K or briefer I J| K.  The main results of this paper answer this question via a characterization based on a directed graph that encodes the sparsity assumptions about the map f.










Before proceeding, we firm up the framework.

If f is Lipschitz-continuous, the SDE in (<ref>), initialized at a random vector 𝕏(0) that is independent of the driving Brownian motion, has a unique but not necessarily stationary solution.



Existence and uniqueness of a stationary solution may be guaranteed in the framework provided by <cit.>.  To this end, we assume that f has component functions f_i that are locally m-integrable for m>, and that f satisfies a Lyapunov condition. Referring to the entries of the diagonal diffusion matrix C =  (c_ij), define the  adjoint Fokker–Planck operator 

    ℒ = ∑_i=1^1/2c_ii∂_i^2 + f_i ∂_i.

 The Lyapunov condition then requires existence of a non-negative function U ∈ C^2(ℝ^), the Lyapunov function, fulfilling
 
    lim_x→∞ U(x) = ∞,  and lim sup_x→∞ℒU(x) < 0.








 
In the sequel, we use graphs to represent the sparsity structure of a drift map f.
Specifically, we consider directed graphs = (V, E) whose vertex set V= [] indexes the coordinate processes of 𝕏.
For all graphs considered in the paper, we  tacitly assume that the edge set includes all self-loops i→ i.

Let _(i) = {j ∈[]: (j,i) ∈ E} denote the set of parents of node i∈ V in graph .  By our convention on self-loops, i∈_(i).


We call a drift map f:ℝ^→ℝ^ -compatible  if it is Lipschitz-continuous, satisfies the above local integrability and Lyapunov conditions, and has its sparsity pattern given by , that is, each coordinate function f_i:ℝ^→ℝ only depends the values x__(i) of input coordinates indexed by _(i), i∈[].




We now introduce our main theorem.  If there is a directed path from node i to node j in the directed graph , then i is an ancestor of j.  Write _(i) for the set of all ancestors of i.  Note that i∈_(i).  
Given , we construct an auxiliary graph =(V,Ê) that we call the trek graph.  It is obtained by including a bidirected edge i↔ j in Ê if _(i)∩_(j)≠∅. 


    
    Let =(V,E) be a directed graph on V=[], and let  be its trek graph.  Let I,J,K⊆[] be three disjoint index sets.  Then the following two statements are equivalent: 
    
        
  (i) The set I is separated from J by V ∖ (I ∪ J ∪ K) in the trek graph .
        
  (ii) The conditional independence X_I  X_J | X_K holds in any random vector X that is distributed according to a stationary distribution P_f,C given by a -compatible drift map f:ℝ^→ℝ^ and a positive definite diagonal matrix C∈ℝ^×. 
    


    




   Consider the graph  depicted in <Ref>, and let X be a random vector following a stationary distribution  given by a -compatible drift map and a positive definite diagonal diffusion matrix. The graph  has the ancestor sets,
   
   _(1)   ={1},      _(2)   ={1,2},       _(3)   ={1,2,3,4},      _(4)   ={4}.
   
   As only the pairs of nodes (1,4) and (2,4) do not have a common ancestor, the trek graph  is missing precisely the edges 1 ↔ 4 and 2 ↔ 4; see <Ref>.  For subsets of nodes I, J, and S, write I⊥_ J| S if I and J are separated by S in  (i.e., every path from a node i∈ I to a node j∈ J intersects S).  Then <Ref> implies the following marginal and conditional independencies:
   
       {1,2}   ⊥_{4}|{3}              X_{1,2}    X_4,

       {1}   ⊥_{4}|{2,3}              X_1    X_4,

       {2}   ⊥_{4}|{1,3}              X_2    X_4,

       {1}   ⊥_{4}|{3}              X_1    X_4|X_2,

       {2}   ⊥_{4}|{3}              X_2    X_4|X_1.
   
   Here, the first listed separation implies the other separations graphically, and the first listed marginal independence implies the other independencies probabilistically by the semi-graphoid axioms of conditional independence <cit.>. 

    <Ref> further states that no marginal or conditional independencies  other than the ones listed above hold for all considered stationary distributions. For instance, <Ref> implies existence of a -compatible drift map f and a positive definite diagonal diffusion matrix C such that 
    
    X_1 X_4 | X_3

    for X ∼ P_f,C. We demonstrate how to construct such distributions in <Ref>.


The proof of <Ref> will be developed in the subsequent sections.  The fact that separation in the trek graph implies conditional independence will be derived with the help of the Fokker–Planck equation that characterizes the density of P_f,C (<Ref>). In constructing counterexamples that exhibit conditional dependence when separation fails, we strive hard to structure them as simple as possible.  Specifically, our counterexamples take f to be linear, which gives an Ornstein–Uhlenbeck process for which we let the diffusion matrix be a multiple of the identity (<Ref>). 


Stationary diffusion processes have emerged as a new framework for statistical analysis of causal systems.  Indeed, in many applications it is difficult if not impossible to observe a system more than once; e.g., in biology, measurement may destroy the observed collection of cells. 
The available data then constitute a sample of independent observations, for which the new paradigm retains an implicit temporal perspective by considering one-time snapshots of independent stationary dynamic processes. This approach was pioneered using stationary Ornstein–Uhlenbeck processes, resulting in Gaussian models termed Lyapunov models <cit.>.  Research progresses on issues like identifiability of drift parameters, statistical estimation, and new applications <cit.>.  Moreover, extensions to non-linear diffusions were proposed in <cit.>.  <Ref> advances this line of research via a characterization of conditional independence  that can be used to disprove parameter identifiability <cit.>. It also reveals that even for general drift maps, the trek graph  may be estimated by  learning a marginal independence graph <cit.>.



Working with stationary distributions of sparse multivariate diffusion processes has particular appeal when modeling causal systems with feedback loops, which present mathematical and interpretational challenges for the classical structural equation models (SEMs) <cit.>. For the latter, the well-known d-separation criterion yields a graphical characterization of conditional independence relations <cit.>. In our <Ref>, a marginal independence X_I X_J (with conditioning set K=∅) arises from the absence of common ancestors of nodes in I and nodes in J, which is precisely the same phenomenon as in d-separation. However, as will be evident from our later proof, <Ref> shows that for stationary diffusions the marginal independencies are the only source of independence relations, which radically differs from the case of d-separation.
















The remainder of the paper is organized as follows.

<Ref> treats the characterization of marginal independence and describes how to infer the conditional independence relations implied by these marginal independencies by means of a connected set Markov property for the trek graph.

In <Ref>, we set up notation, formally introduce the Lyapunov model and review some facts on Gaussian distributions. 

In <Ref>, we prove the main result by showing that all conditional independence relations that hold in the Lyapunov model originate from marginal independencies.  To this end, we reduce the problem to graphs obtained from 1-clique sums of treks. We then use a combination of algebraic and analytic methods to establish the result in this simpler case.





§ MARGINAL INDEPENDENCE AND IMPLIED CONDITIONAL INDEPENDENCIES



This section proves one direction in <Ref>, namely, that V ∖ (I ∪ J ∪ K) separating I and J in the trek graph  implies the conditional independence X_I  X_J | X_K.  We start with the case K=∅, i.e., marginal independence statements. Then, we show that these statements imply all claimed conditional independence statements. Before giving the proofs, we introduce graphical concepts, as used in this as well as later sections of the paper.



 §.§ Graphs
 

For a directed graph = (V, E) with edge set E ⊆ V × V, we write i → j if (i,j) ∈ E.  We denote the set of nodes or edges of a specific graph  by V() or E(), respectively.  The set _(i) = {j ∈ V: j → i} collects the parents of a node i∈ V, and _(i) = {j ∈ V: i → j} is its set of children. 
A directed path is a sequence of 

nodes (i_1, …, i_m) such that i_l → i_l+1∈ E for 1≤ l<m, and m-1 is the length of this path. A path of length 0 consists of a single node.
As defined in the introduction, _(i) is the set of ancestors of i, i.e., the nodes j from which there is a directed path to i.
This definition is conjunctively applied to sets, so _(I) = ⋃_i ∈ I_(i). The set of descendants of a node i is denoted _(i) and consists of all nodes j such that there is a directed path from i to j. Typically, it is clear which graph  is concerned, and we  drop the subscript, writing (i)≡_(i) and, similarly,  (i) and (i). The respective sets where the node itself is excluded are denoted in lowercase, e.g., (i)=(i)∖{i} and (i)=(i)∖{i}.

Combining two directed paths yields a trek: An (ℓ,r)-trek between nodes i_ℓ and j_r is a sequence of 

nodes (i_ℓ, …, i_1, t, j_1, …, j_r) such that both (t, i_1, …, i_ℓ) and (t, j_1, …, j_r) are directed paths.  Node t is the top node of the trek.  If there is a trek between two nodes i and j, then the top node is a common element of (i) and (j).  Thus, the trek graph  from <Ref> has two nodes adjacent precisely if there is a trek between them.

A subset of nodes I ⊆ V is ancestral if (I) = I. A trivial example of an ancestral set is a source node, meaning a node without parents.  
If a directed path starts and ends at the same node, then it forms a directed cycle. If the graph  does not contain any directed cycles, then it is termed a directed acyclic graph (DAG).  The vertex set of a DAG may be ordered topologically as (i_1, …, i_n) such that for l ≠ k, i_l∈(i_k) implies that l<k.










 §.§ Marginal Independence


The first result treats the implication (i)⇒(ii) in <Ref> when the conditioning set K is empty, in which case the independence is marginal.

 
Let 𝒢=(V,E) be a directed graph on V=[]. If f is 𝒢-compatible, then X ∼ P_f, C satisfies X_I  X_J for all sets of nodes I, J ⊆ V without trek between them.

This theorem generalizes an observation concerning Ornstein–Uhlenbeck processes made in <cit.>, and we prove it using the Fokker–Planck equation.  To this end, we draw on the next lemma, which follows from combining Theorem A and Proposition 2.1 in <cit.>.  Here, C_0^∞(ℝ^) is the set of smooth functions with compact support.

 If the Lyapunov condition is fulfilled, then (<ref>) has a unique stationary solution admitting a density p. This density is the unique density fulfilling the Fokker–Planck equation
  
    ∫_ℝ^ℒϕ(x)p(x) dx = 0  for all ϕ∈ C_0^∞(ℝ^) .
 
 



 
By <Ref>, the stationary distribution P_f, C has a density p.
We begin by showing that for any ancestral set A⊆ V, the marginal density p_A of X_A satisfies the Fokker–Planck equation for the SDE restricted to A. Let A^ = V ∖ A be the set of all remaining nodes. Fix a function ϕ∈ C_0^∞(ℝ^) with ϕ(x) only depending on x_A. Then the Fokker–Planck equation for the overall density p yields that

    0 =  ∫_ℝ^ℒϕ(x)p(x) dx 
     =∫_ℝ^( ∑_i=1^1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) p(x) dx 
    
     = ∫_ℝ^( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) p(x) dx
 
because ∂_i ϕ(x) ≡ 0 for i ∉ A as ϕ only depends on x_A. Now, the ancestrality of A implies that f_i for i∈ A only depends on x_A.  Rewriting the above integral, we obtain, by Fubini,

    0 
     
     
      =    ∫_ℝ^A∫_ℝ^A^( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) p(x) dx_A^ dx_A 
    
      =    ∫_ℝ^A( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x) + f_i(x) ∂_i ϕ(x) ) ∫_ℝ^A^p(x) dx_A^ dx_A 
    
      =    ∫_ℝ^A( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) ) p_A(x_A)  dx_A,
 
  which is the Fokker–Planck equation for a restricted SDE of the form
  
    d𝕏_A(t) = f_A(𝕏_A(t))  dt + D_A, A d𝕎(t)

    with D_A,AD_A,A^T=C_A,A.
Applying this result to A=(I) ∪(J) gives a Fokker–Planck equation for the marginal density of  X_(I) ∪(J). To show the independence statement, we show that the product of marginal densities p_(I) p_(J) fulfills the same Fokker–Planck equation. Let Ĩ = (I), J̃ = (J), and ϕ∈ C_0^∞(ℝ^A). Since there is no trek between I and J, the sets Ĩ and J̃ are disjoint and partition A=Ĩ∪J̃. Therefore,

    ∫_ℝ^A( ∑_i∈ A1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A 
     =   ∫_ℝ^A( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A 
       +∫_ℝ^A( ∑_i∈J̃1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A.

We rewrite the first integral as

    ∫_ℝ^A( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_A 
     =   ∫_ℝ^J̃∫_ℝ^Ĩ( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )(p_Ĩ p_J̃)(x_A) dx_Ĩ dx_J̃
     
     =   ∫_ℝ^J̃∫_ℝ^Ĩ( ∑_i∈Ĩ1/2c_ii∂_i^2 ϕ(x_A) + f_i(x_A) ∂_i ϕ(x_A) )p_Ĩ (x_Ĩ)  dx_Ĩ  p_J̃(x_J̃)  dx_J̃.
 
  The Fokker–Planck equation for Ĩ, with test function x_Ĩ↦ϕ(x_Ĩ,x_J̃) for fixed x_J̃, yields that the inner integral vanishes. Hence, the overall integral vanishes as well. Reversing the roles of I and J shows that also the second summand in (<ref>) is zero.  In conclusion, p_(I) p_(J) fulfills the Fokker–Planck equation for p_(I)∪(J), i.e., p_(I) p_(J) is a density of X_(I)∪(J).  Consequently, X_(I) X_(J) and the decomposition property of conditional independence implies our claim that X_I X_J because I⊆(I) and J⊆(J).




 §.§ Implied Conditional Independencies












   




Let =(V,Ê) be the trek graph associated to the given directed graph .  By definition, (i,j)∈Ê if and only if there is a trek between nodes i and j.  Observe that (i,j) ∈Ê if and only if (j,i) ∈Ê, and, reflecting the structure of treks, we draw the edge bidirected as i↔ j.  <Ref> clarifies that the trek graph encodes marginal independencies, in particular,

    i↔ j ∉Ê X_i X_j

in random vectors X∼ P_f,C with f a -compatible drift map.
A more refined interpretation of the trek graph allows us to derive all conditional independencies that are implied by the marginal statements from <Ref>.   Recall the notation I⊥_ J| S  from <Ref>, which indicates that the subsets of nodes I and J are separated by subset S in .







Let =(V,E) be a directed graph on V=[], and consider a random vector X∼ P_f,C, where f:ℝ^→ℝ^ is a -compatible drift map and C∈ℝ^× is positive definite and diagonal.  Let I,J,K⊆[] be three disjoint index sets, and let  be the trek graph of .   Then 

    I⊥_ J| V ∖ (I ∪ J ∪ K)     
    X_I  X_J | X_K.






In the literature on probabilistic graphical models, a random vector X that satisfies (<ref>) is said to satisfy the pairwise Markov property relative to the bidirected graph  <cit.>.  However, <Ref> shows that a richer set of marginal independencies hold.  Call a subset A⊆ V connected in  if any two vertices i, j ∈ A are connected by a path in  all of whose vertices are in A (i.e., A induces a connected subgraph of ).  Let 
Sp_(A) be the set of all nodes that are in A or are adjacent to an element of A via an edge of .  Then <Ref> implies that X satisfies the connected set Markov property relative to , which requires that 

    X_A  X_V ∖Sp_(A)  ∀  A⊂ V,  non-empty and connected in .

By general properties of conditional independence, the connected set Markov property is equivalent to the global Markov property for bidirected graphs <cit.>.  The global Markov property states that 

    I⊥_ J| V ∖ (I ∪ J ∪ K)     
    X_I  X_J | X_K,

as was the claim of our theorem.



We illustrate the theorem in another example.




Let  be the bidirected graph from <Ref>.  The connected set Markov property requires

    1{3,4},  {1,2} 4.

As 1 is separated from 4 by 3 in , the global Markov property additionally states 1  4 | 2. The other conditional (and not marginal) independencies required by the global Markov property are

    1 3 | 4,   1 4 | 3,   2 4 | 1.




In the next sections, we show that <Ref> already captures all conditional independencies that hold for general -compatible drift. To this end, we consider Ornstein–Uhlenbeck processes and the graphical Lyapunov models obtained from their stationary distributions.













§ ORNSTEIN–UHLENBECK PROCESSES AND LYAPUNOV MODELS



This section briefly reviews Ornstein–Uhlenbeck processes and the associated graphical continuous Lyapunov models. We also recall basic facts regarding Gaussian conditional independence that will be used in <Ref>.



 §.§ Ornstein–Uhlenbeck Processes



The Ornstein–Uhlenbeck process is the only non-trivial time-homogeneous Markov process that is both Gaussian and stationary <cit.>.  In the -dimensional case, it is defined by the SDE

    d𝕏(t) = M (𝕏(t)-μ)  dt + D  d𝕎(t)

with an invertible drift matrix M ∈^× encoding the interactions among the coordinates of 𝕏(t). 
The affine linear drift term introduces a restoring force towards the mean vector μ.

Again we only consider the case with D and the diffusion matrix C=D D^T diagonal.


Evidently, (<ref>) is a special case of
(<ref>) with f being an affine
function given by M and μ.  In this case, the Lyapunov condition
on f is satisfied if M is a stable matrix, that is, the
real parts of the eigenvalues of M are negative.  For M stable,
𝕏(t) admits a unique stationary distribution that is Gaussian with mean μ <cit.>.  The positive definite covariance matrix of the distribution is the unique solution Σ≡Σ(M,C) of the continuous Lyapunov equation

    MΣ + Σ[-0]M = -C.






We note that the solution Σ is invariant under joint rescaling of M and C. 

Given our interest in conditional independence structure, we restrict
ourselves, without loss of generality, to stationary distributions of
Ornstein–Uhlenbeck processes with mean μ = 0. For a directed
graph , the class of Gaussian distributions of interest 
naturally corresponds to the set of covariance matrices Σ
that can be achieved using stable matrices M for which the linear
map x ↦ Mx is -compatible.  In other words, we consider
stable matrices M whose support (i.e., non-zero entries) is
contained in the edge set E(). 






    Let = (V, E) be a directed graph on V = []. Let ^E be the set of matrices M = (m_ij) ∈^× such that m_ji = 0 if i → j ∉ E and denote by _ the cone of × positive definite matrices. Then for any fixed C ∈_, the (graphical continuous) Lyapunov model of  is the family of all multivariate Gaussian distributions (0,Σ) with covariance matrix Σ in the set
    
    _,C = {Σ∈_ :  M Σ + Σ M^T + C = 0  for a stable matrix  M ∈^E}.


In the sequel, we identify the model with its set  of covariance matrices and refer to _,C as the Lyapunov model. We denote the entries of a covariance matrix Σ by σ_ij. 


We denote the -dimensional identity matrix by _ or equivalently by _V for a set V with |V| =. We will typically fix C = 2 _ and write _ = _, 2 _. This assumption guarantees that _∈_, C which is a desirable property for any Gaussian model. While the drift matrix M is typically thought of as the weighted adjacency matrix of , we will usually suppress the self-loops in subsequent drawings of graphs for simplicity. 







 §.§ Conditional Independence in Multivariate Gaussians






Consider a symmetric matrix Σ with rows and columns indexed by a finite set V. For I,J⊆ V, let Σ_I,J denote the submatrix with rows indexed by I and columns by J. Submatrices of the form Σ_K Σ_K,K are principal. 

Especially in subscripts, we will often abbreviate a union I ∪ J of index sets to IJ. A single index i is used interchangeably with its singleton set {i} in these conventions.
So, iK{i}∪ K and K ∖ i K ∖{i}.


The starting point of our subsequent study of conditional independence is the following collection of results. Proofs can be found, for instance, in 4.1 of <cit.>.














 
Let Σ∈_n.
The conditional independence (CI) statement I  J | K holds in the Gaussian random vector X ∼(μ, Σ) if and only if submatrix
Σ_IK, JK has rank |K|. 


Since Gaussian distributions satisfy the compositional graphoid axioms, the CI statement I  J | K holds if and only if i  j | K holds for all i ∈ I and j ∈ J <cit.>. Thus, to characterize the set of CI statements that a Gaussian distribution satisfies, it suffices to characterize all statements of the form i  j | K. This is summarized with the following corollary, which combines this fact with the previous theorem. 




For a Gaussian random vector X∼(μ, Σ) with Σ∈_n, the following statements are equivalent:


  * I  J | K,

  * i  j | K for all i ∈ I and j ∈ J, and

  * (Σ_iK, jK) ≡ |Σ_iK, jK| = 0 for all i ∈ I and j ∈ J.



This characterization of conditional independence for multivariate Gaussians will be used extensively in <Ref>.






























§ CONDITIONAL INDEPENDENCE IN LYAPUNOV MODELS



Our focus is now on the 

Lyapunov model of a directed graph = (V,E), which is given by the set of covariance matrices 

    _ = {Σ∈_ :  M Σ + Σ M^T + 2 _ = 0   for a stable matrix  M ∈^E}.


<ref> applies to _ 
and shows that the absence of a trek in  leads to a marginal independence, and <ref> yields the implied 

additional conditional independence (CI) statements for _.
The aim of this section is to prove that the resulting collection of CI statements is complete, by showing that for any other CI statement, there is a distribution Σ∈_ which violates it.
More precisely, our main result is as follows.






Let  be a directed graph with trek graph . Then the Lyapunov model _ is Markov-perfect to the marginal independence graph . In other words, i  j | K holds for all Σ∈_ if and only if  V ∖ ({i ,j }∪ K) separates i and j in . 


The proof of this theorem will be completed at the end of this section. As noted above, the direction to be proven is the fact that a CI statement holds for all Σ∈_ only if we have graphical separation.




 §.§ Outline of the Proof



We will prove the contrapositive, 
i.e., we will show that if V ∖ ({i ,j }∪ K) does not separate i and j in , then there exists a matrix Σ∈_ such that i j | K.  To do this, we first clarify that we may reduce the problem to studies of suitable subgraphs, and we introduce a trek rule which gives a polynomial parameterization of a submodel of _ in terms of the entries of M (<ref>). We then show how to apply this restricted trek rule to the case where  is a single trek in order to produce covariance matrices Σ in _ such that the endpoints of the trek are conditionally dependent given all other nodes on the trek 
Via a trick of creating a limiting scenario of perfect correlation, we  extend this result to more general conditioning sets comprised of arbitrary subsets of the other nodes on the trek (<ref>).

Finally, to complete the proof, we extend the previous results from a single trek to zig-zag graphs, which correspond to paths that are connecting for the global Markov property described in the proof of <ref> (<ref>).  We note that these zig-zag graphs also appear in the description of the d-separation criterion for structural equation models; cf. <cit.>. 



 §.§ Subgraphs and a Restricted Trek Rule



The following lemma clarifies how to pass to subgraphs for the construction of counterexamples to CI statements.


 
Let = (V, E) be a directed graph, and let ' = (V', E') be a subgraph of . 
If i j | K' on _', then i j | K on _ for all K ⊆ V with K ∩ V' = K'.



The hypothesis that i j | K' means that there exists Σ' ∈_' for which |Σ'_iK',jK'| ≠ 0. Let M' be the corresponding parameter matrix, i.e., M'Σ' + Σ' M'^T + 2 _V' = 0. Denote V” = V ∖ V'. Then the matrix equation with blocks indexed by V' and V”

    [   M'    0;    0 -_V” ]_ M[  Σ'   0;   0 _V” ]_Σ + 
      [  Σ'   0;   0 _V” ]_= Σ[ M'^T    0;    0 -_V” ]_= M^T +
      [ 2 _V'     0;     0 2 _V” ]_= 2 _V = 0

shows that Σ∈_. Since K ∩ V' = K', it follows that |Σ_iK,jK| = |Σ'_iK',jK'| ≠ 0.


The previous lemma will be used throughout this section to reduce the problem of determining whether a conditional dependence holds to a minimal subgraph which exhibits it. The next proposition is then our major tool to study the resulting minimal subgraphs.




Let = (V, E) be a DAG, and consider the slice of the model _ obtained by restricting the drift matrix M=(m_ij) to have diagonal entries m_ii = -12ζ for all i ∈ V. Then the following trek rule holds for the entries of the covariance Σ=(σ_ij) associated to M:

    σ_ij = ∑_T : (ℓ, r)-trek
    from i to j 2 ζ^ℓ+r+1ℓ+rℓ m_T,

where the trek monomial m_T associated to a trek T is given by m_T = ∏_e ∈ Tm_e, i.e., the product over all the edges e in that trek.




The Lyapunov equation MΣ + Σ M^T = -C is linear in Σ, and 












the ij-th element of the matrix equation reads

    (m_ii + m_jj) σ_ij + ∑_l ∈(j) m_jlσ_li + ∑_k ∈(i) m_ikσ_kj = -C_ij.

Note that the coefficient m_ii + m_jj of σ_ij is always -1ζ. Solving for σ_ij, we obtain

    σ_ij = ζ·(C_ij + ∑_l ∈(j) m_jlσ_li + ∑_k ∈(i) m_ikσ_kj).

These relations motivate a proof of (<ref>) by induction. 

We fix a topological ordering of the DAG  and extend it to a lexicographic ordering on the entries of Σ via σ_ij < σ_kl if and only if j < l or j = l and i < k. We perform induction with respect to this ordering, noting that the right-hand side of (<ref>) contains only Σ-entries which are strictly smaller than σ_ij on the left-hand side.

The smallest Σ-entries are σ_ii where i is a source node. In this case σ_ii = ζ C_ii = 2ζ follows immediately from the Lyapunov equation for our choice of C = 2_V. This verifies the claim as there is only the empty trek on i. <Ref> or equation (<ref>) can be used to see that if i and k are distinct source nodes (having in particular no treks between them), then σ_ki = 0. Now let j be a node all of whose parents are source nodes; the edges into such nodes j form the next smallest class of Σ-entries in the lexicographic ordering. The Lyapunov equation simplifies to σ_ij = ζ∑_k ∈(j) m_jkσ_ki = ζ m_jiσ_ii = 2 ζ^2 m_ji which corresponds to the unique trek between i and j of length ℓ+r = 1. These two cases prove the initial induction hypothesis.

Fix any two (not necessarily distinct) vertices i, j and a non-empty (ℓ,r)-trek T between them. This trek decomposes into a unique (ℓ,r-1)-trek T_1 from i to l, for l ∈(j), and the edge l → j. It also decomposes uniquely into an (ℓ-1,r)-trek T_2 from k to j, for k ∈(i), together with the edge k → i. By (<ref>), the coefficient of m_T in σ_ij is thus ζ times the sum of the coefficients of m_T_1 in σ_il and m_T_2 in σ_kj. Using the induction hypothesis (since σ_il and σ_kj are both less than σ_ij) gives

    coefficient of m_T = ζ·(2 ζ^ℓ+rℓ+r-1ℓ + 2 ζ^ℓ+rℓ+r-1ℓ-1) = 2 ζ^ℓ+r+1ℓ+rℓ,

as claimed. This immediately proves the induction step when i ≠ j. In case i = j, there is also the empty trek from i to i to account for and its trek term arises as required from (<ref>) via C_ii = 2.







Consider the graph in <Ref>. Using the trek rule formulation for σ_ij from <ref>, we get 

    σ_11   =    2ζ, 
    σ_22   =    4ζ^3m_21^2 +2ζ, 
    σ_33   =    4ζ^3m_31^2 +2ζ, 
    ⋮
    σ_34   =    6ζ^4m_21m_31m_42 + 6ζ^4m_31^2m_43 + 2ζ^2m_43.

For instance, observe that there are two treks from the vertex 2 to itself, the empty trek (corresponding to the monomial 2ζ for σ_22) and the trek 2← 1 → 2 (4ζ^3m_21^2). Similarly, there are three treks between the vertices 3 and 4, which are the edge 3→ 4 (2ζ^2m_43), the trek 2← 1 → 3 → 4 (6ζ^4m_21m_31m_42), and 3← 1 → 3 → 4 (6ζ^4m_31^2m_43). For the last trek 3← 1 → 3 → 4, we have ℓ=1 and r=2, which gives us the coefficient 6ζ^4.



This trek rule is reminiscent of the original trek rule for linear structural equation models <cit.>. However, now each trek monomial is weighted by a binomial coefficient. While the trek rule we present here only parameterizes a proper subset of _ and is less general than that of <cit.>, it provides an explicit rational formula for the entries of Σ in terms of the parameters M which makes it significantly easier to construct explicit covariance matrices in _ with the desired dependence properties. We remark that the very recent preprint of <cit.> gives a generalization of <Ref>.



 §.§ Conditional Independence on Treks



We now turn to the case where the considered graph is a single trek, for which we show that the associated Lyapunov model does not satisfy any CI relations.
Before proceeding, we note that if  is a trek from node 1 to  with top node t, then  a drift matrix M∈ℝ^E() 
    
    has zero-pattern
    
    M = ([   d_1     *                              ;           ⋱     ⋱                        ;             d_t-1     *                  ;                     d_t                  ;                       * d_t+1            ;                             ⋱     ⋱      ;                                   *    d_;       ])

    with diagonal entries d_1,…,d_ and the symbol *
    indicating arbitrary values.  Such matrices are stable if and only
    if the diagonal entries are negative.
    





Let  be an (ℓ,r) trek between i and j. Let K = V() ∖{i, j} be the remaining vertices. Then i j | K in _. 


The proof of this proposition and its several supporting lemmas can be found in <ref>. The following example provides an explicit family of covariance matrices in the Lyapunov model for a small trek which exhibit the conditional dependence stated in <Ref>. The proof of the previous proposition generalizes this family of covariance matrices to larger treks and shows that the conditional dependence still holds by inductively computing certain entries in the adjoint matrix of Σ_iK, jK. The Schur complement expansion of |Σ_iK, jK| with respect to the principal minor Σ_K,K can then be used to show that |Σ_iK, jK| ≠ 0 by inductively computing the lowest degree term which appears in |Σ_iK, jK| when it is viewed as a polynomial in ζ. 






Let  be the trek as shown in <Ref>. Setting every edge parameter m_ji to 1 for all i→ j ∈ E(),

we get the following covariance matrix from the trek rule which lies in the model:

    Σ=[            2ζ          2ζ^2          2ζ^2          2ζ^3;          2ζ^2       4ζ^3+2ζ          4ζ^3          6ζ^4;          2ζ^2          4ζ^3       4ζ^3+2ζ     6ζ^4+2ζ^2;          2ζ^3          6ζ^4     6ζ^4+2ζ^2 12ζ^5+4ζ^3+2ζ ].

Observe that the submatrix Σ_{1,2,3},{2,3,4} has determinant -8ζ^6, which is non-zero for ζ≠0, implying that 2 4 |{1,3} in the model _.


We now introduce a technique to extend the previous proposition to all conditioning sets K ⊆ V() ∖{i, j}.  In Lyapunov models with fixed diffusion matrix C=2 _V, we cannot simply pass to marginal distributions as, in 

general, removing a row and corresponding column in a covariance matrix Σ∈_ does not yield

a distribution in the Lyapunov model over the shortened trek in which the removed node is skipped.

Instead, we provide a construction in which the random variables indexed by nodes in V() ∖ K become perfectly correlated with random variables indexed by elements of K.  This construction then allows us to argue that the counterexample from <ref> can be `lifted' to counterexamples on longer treks between i and j that contain additional nodes not in the conditioning set K.







To introduce the construction, we recall the following notation. 
For a node k for which (k) is a singleton set, we identify (k) with its single element, i.e., the unique node j with j→ k. Similarly,  we identify (k) with the unique node j such that k→ j.  Moreover, for k ∈, we define a map that shifts indices as

    ψ_-k: 
        ∖{k}→,    
        x ↦
            x    if  x < k; 
    
            x-1    if  x > k.


Now, consider a trek  between i and j with top node t and of length |V()|≥ 4. If |(t)| = 2, let c_l and c_r be the child nodes on the left and on the right side of the trek, respectively. Let k ∈ V()∖{i,j} be a node on the trek, and consider the subset of nodes V_-k = V()∖{k} obtained by removing k.  Define a shorter trek 

    _-k = (V_-k,E_-k)


by constructing the edge set E_-k as follows.
If k≠t,  replace the 2-path (k) → k →(k) by the single edge (k)→(k). If k=t, the trek  has two non-trivial sides as k∉{i,j}.

In this case, if the right side of  has length 1, we replace c_l ← k → c_r by c_l→ c_r. If k = t and the right side consists of more than one edge, we replace c_l ← k → c_r by c_l← c_r. The construction is depicted in <Ref>.
In all cases, the resulting graph is shorter by one node, as the k-th node is removed. Note that a similar definition is possible by adding the reverse edge c_l → c_r in the case where k = t and both sides of the trek consist of more than one edge. The distinction between the second and third case is necessary to keep the trek structure when removing a top node from a trek where one side consists of exactly one edge. Otherwise, the resulting graph could be a directed path, e.g., _-k could be i  c_l  c_r in <Ref>. 
In the case that the nodes V() = [] are numbered consecutively from left to right, we implicitly apply ψ_-k to the elements of V_-k and the start- and endpoint of each edge in E_-k so that we have V_-k = [-1] and E_-k⊆ [-1] × [-1].



Given a matrix Σ, we write 

    Σ_-k,-k = Σ_[]∖{k},[]∖{k}

for the submatrix obtained by removing the kth row and column of Σ.
Given a set of matrices ⊆^×, we denote its topological closure by .

 
    Let  be a trek between i and j containing more than 3 nodes.  Let t be the top node of , and let k ∉{i,t,j} be a further node on .  Then for every matrix Σ^* ∈__-k, there exists a matrix Σ∈_ such that 
    
    Σ_-k,-k = Σ_-(k),-(k) = Σ^*.




    Let ≥ 4 be the number of nodes on the trek .
    Without loss of generality, assume that the nodes on  are numbered consecutively from left to right, i.e., i = 1, j =, and k ∈{2,…,-1}∖{t}. 

    From Σ^* ∈__-k, we are to construct a suitable
    larger covariance matrix Σ∈_.  To do this,
    we make the
    new node at index k a copy of its parent on the
    trek by duplicating the respective row and column from Σ^*. In the resulting distribution,
    
    
    k and (k) are perfectly correlated. 
    Note that (k) = {k+1} for k < t, and (k)={k-1} for k > t. Thus, in the first case, the kth and k+1st row and column of Σ are duplicates of each other, while in the second case, the kth and k-1st row and column of Σ are duplicates of each other. The proof in both cases is analogous as the drift matrix of a single trek is defined in a symmetric way around its top node as seen in (<ref>). We only give the proof for the second case, so assume k > t.
    Furthermore, let U:={1,…,k-1},
    W:={k+1,…,}, and write ψ≡ψ_-k for
    the map from (<ref>). 
    Then, ψ(U) = U and ψ(W) = {k, …, -1}.
    
    We note that the case of a directed path, i.e., a trek with
    i=t=1, is included here as k > 1 by assumption.  

    Given Σ^* ∈__-k, there exists a stable matrix M^* ∈^(-1)×(-1) such that the Lyapunov equation 
    
    M^* Σ^* + Σ^* (M^*)^T + 2_-1 = 0

    is fulfilled.
    In the following, we show that our construction of inserting a
    duplicate of the row and column of Σ^* at index (k) in
     indeed yields a matrix Σ∈_.

    

    To achieve this, we construct a sequence of drift matrices M^(m) as follows. First, partition
    
    Σ^* = 
        ([      2*Σ^*_U 2*Σ^*_U,ψ(W);                          ; 2*Σ^*_ψ(W),U   2*Σ^*_ψ(W);                          ;              ])

    as well as 
    
    M^* = 
        ([      2*M^*_U          2*0;                          ; 2*M^*_ψ(W),U   2*M^*_ψ(W);                          ;              ]),
          where   
        M^*_ψ(W),U = 
        ([ 0    ⋯    0   m^*_k,k-1;           0     0 
    ⋮
     0;             ]).

    Considering (<ref>) block-wise results in the following four equations:
    
        
  * M^*_UΣ^*_U + Σ^*_U[-0]M^*_U + 2_k-1 = 0, 
        
  * M^*_UΣ^*_U,ψ(W) + Σ^*_U,ψ(W)M^*_ψ(W) + Σ^*_UM^*_ψ(W),U = 0, 
        
  * M^*_ψ(W)Σ^*_ψ(W) + Σ^*_ψ(W)M^*_ψ(W) + 2I_-k + M^*_ψ(W),UΣ^*_U,ψ(W) + Σ^*_ψ(W),UM^*_ψ(W),U = 0, 
        
  * M^*_ψ(W)Σ^*_ψ(W),U + Σ^*_ψ(W),U[-0]M^*_U + M^*_ψ(W),UΣ^*_U = 0, 
    
    where equations <ref> and <ref> are transposes of each other.
    Given the stable matrix M^*, these equations are uniquely fulfilled by the corresponding submatrices of Σ^*.
    
    Now, let m ∈_>0 and define
    
    M^(m) = 
        ([   2c|2*M^*_U          2*0        2c2*0;          2c|                        2c;            0            m           -m          2c0;       2c|2*0    m^*_k,k-1 2c2*M^*_ψ(W);          2c|            0           2c;              ]) ∈^×.

    Further, let
    
    Σ^(m) = 
        ([   2c|2*Σ^(m)_U    2*Σ^(m)_U,k  2c2*Σ^(m)_U,W;            2c|                            2c;   2c|Σ^(m)_k,U      Σ^(m)_k,k    2cΣ^(m)_k,W; 2c|2*Σ^(m)_W,U    2*Σ^(m)_W,k    2c2*Σ^(m)_W;            2c|                            2c;                ]) ∈^×

    be the solution of the resulting Lyapunov equation 
    
    M^(m)Σ^(m) + Σ^(m) (M^(m))^T + 2_ = 0

    induced by M^(m) for every m ∈_>0. 
    
    Due to the symmetry of the Lyapunov equation, any unique solution of (<ref>) is symmetric. Since M^(m) is stable, the solution Σ^(m) is unique and therefore symmetric, so we only have to solve six matrix equations that arise by block-matrix multiplication, denoted by letters in brackets as follows:
    
    M^(m)Σ^(m) + Σ^(m) (M^(m))^T + 2_ = 0 ≡([ 2c|2*(A)    2*(B)  2c2*(C);      2c|                2c;      2c|      (D)    2c(E);    2c|2*       2*  2c2*(F);      2c|                2c;          ]).


    The upper left block Σ^(m)_U of the partitioned matrix
    Σ^(m) is constant as it is the unique solution of
    equation <ref>. Proceeding recursively, the other
    blocks of Σ^(m) can be shown to converge for m →∞ as well, with limits given by the relevant parts of Σ^*. The details of these calculations can be found in <Ref>.
    Combining the results, we have
    
    lim_m →∞Σ^(m) = 
        ([      2c|2*Σ^*_U     2*Σ^*_U,(k)  2c2*Σ^*_U,ψ(W);             2c|                              2c;    2c|Σ^*_(k),U     Σ^*_(k),(k)  2cΣ^*_(k),ψ(W); 2c|2*Σ^*_ψ(W),U  2*Σ^*_ψ(W),(k)    2c2*Σ^*_ψ(W);             2c|                              2c;                 ]) ∈_,

as desired.


 
    Let  be a trek with endpoints i and j. Let ∅≠ K ⊊ V()∖{i,j} and let k ∉{i,j}∪ K be a node on the trek. If there is 
    
    Σ^* ∈__-k such that  | Σ^*_ψ_-k(iK),ψ_-k(jK)| ≠ 0,

    then there is 
    
    Σ∈_ such that  |Σ_iK,jK| ≠ 0.




    By continuity of the determinant, it suffices to construct a matrix Σ∈_ such that | Σ_iK,jK| ≠ 0.  Considering the closure of the model is crucial as it allows us to apply <ref> for the construction of Σ.   Then as k ∉{i,j}∪ K, we find that 
    
    | Σ_iK,jK| 
            = | Σ^*_ψ_-k(iK),ψ_-k(jK) |
             ≠ 0.






    
















































    

























 §.§ Conditional Independence in Zig-Zag Graphs




We now introduce the family of graphs that will be the key element needed to complete the proof of <ref>. 


Let _1, …, _p be treks such that for i ∈ [p-1], there exists a node k_i on the right branch of _i which is the endpoint of the left branch of _i+1. A zig-zag graph  (<Ref>) is a directed acyclic graph obtained by performing an iterated 1-clique sum of these treks along the nodes k_i, i.e.

    = _1 ⊕_k_1_2⊕_k_2⋯_p-1⊕_k_p-1_p.














Let  be a zig-zag graph with endpoints i and j, and let K ⊆ V() ∖{i, j} be a set such that for each collider k ∈ it holds that s ∈ K where s is the unique sink in (k). Then i j | K in the Lyapunov model associated to . 



To see this we use <Ref> inductively on the number p of treks that  is obtained from. If  is simply a trek then the result already holds by <ref>. So suppose that _p is a zig-zag graph with tails obtained from combining p treks and that for any set K' ⊆ V(_p) of the required form it holds that k j | K' where k and j are the endpoints of _p. Let  be a trek from i to s containing a node labelled k which is not on the same branch as i and let _p+1 be the 1-clique sum of  and _p at k. Recall that this is exactly how every zig-zag graph is produced as pictured in <Ref>. Note that if k and i are on the same branch then checking that i j | K on _p+1 naturally reduces to checking i j | K on a smaller zig-zag _p which immediately follows from the induction hypothesis. 


To complete the induction we must show that i j | K where K is of the form K = T ∪ L ∪ K' such that T ⊆ V() ∖{i}∪(k), s ∈ L ⊆(k) and K' ⊆_p is a set such that k j | K'. Lastly, let l ∈ L be the node closest to k. Then observe that after rearranging the rows and columns of Σ_iK, jK it has the block form

    Σ_iK, jK = 
    cccc    jK'                                  l T                              L ∖ l  
    l(ccc)
    l K'              Σ_lK', j K'   ∗   ∗
     
    i T                    0                                   Σ_iT, lT   ∗
    
    L ∖ l     Σ_L ∖ l, jK'   Σ_L ∖ l, lT   Σ_L ∖ l,L ∖ l
     .

Now observe that if we specialize this matrix by setting m_ab = 0 and m_la = 0 for all a,b ∈ L ∖ l then the above block matrix immediately simplifies to

    Σ_iK, jK = 
    cccc    jK'                                  l T                              L ∖ l  
    l(ccc)
    l K'              Σ_lK', j K'   ∗   ∗
     
    i T                    0                                   Σ_iT, lT   ∗
    
    L ∖ l      0                                    0                                   Σ_L ∖ l, L ∖ l
     .

Thus we see that for any such matrix obtained from the submodel corresponding to this specialization, |Σ_iK, jK| = |Σ_lK', j K'|  |Σ_iT, lT|  |Σ_L ∖ l, L ∖ l|. Now observe that |Σ_lK', j K'| ≠ 0 by our induction hypothesis since this corresponds to conditional dependence of a zig-zag with endpoints l and j which is composed of only p treks. Further observe that |Σ_iT, lT| ≠ 0 by <ref>. The last factor is a principal minor and thus is nonzero for any Σ∈__p+1. So any Σ which comes from the above submodel satisfies |Σ_iK, jK| ≠ 0 and thus i j | K for the Lyapunov model _. 
































Consider the zig-zag graph  pictured in <Ref>. Then 3 5 |{1, 4} since for any Σ∈_, it holds that

    Σ_{3,1,4}, {5, 1, 4} = 
    cccc    5                  4                 1 
    l(ccc)
    4      σ_45   σ_44   σ_14
     
    3       0                 σ_34   σ_13
    
    1       0                 σ_14   σ_11
     .

As was shown in the previous proof, this matrix is block upper-triangular with blocks σ_45 and Σ_{3,1}, {4,1}. 
Thus |Σ_{3,1,4}, {5, 1, 4}| = σ_45 |Σ_{3,1}, {4,1}| where the second term in the product is guaranteed to be nonzero for almost all choices of parameters by <ref>. 



We now have all the necessary tools needed to complete the proof of <ref>. 


If V ∖ ({i ,j }∪ K) separates i and j in the marginal independence graph  then <ref> immediately implies that i  j | K holds for all Σ∈_. It remains to show that if i  j | K then V ∖ ({i, j}∪ K) separates i in j in . We instead show the contrapositive: if i and j are connected given V ∖ ({i, j}∪ K) in , then there exists Σ∈_ such that i j | K. We will prove this by constructing a specific Σ using the trek rule described in <ref> and then showing that |Σ_iK, jK| ≠ 0 which by <Ref> implies that the conditional dependence i j | K holds.



If i and j are connected in  given V ∖ ({i ,j }∪ K) then there is path in  from i to j which uses only the vertices in {i, j}∪ K. Without loss of generality, this path is simple, i.e., it does not repeat a vertex. Let i = k_0, k_1, …, k_s, k_s+1 = j be the sequence of vertices on this path where {k_1, …, k_s}⊆ K. Each edge k_p  k_p+1 in  corresponds to a trek in . Even though the path in  is simple, these treks can still intersect in . Suppose that the edges k_p  k_p+1 and k_q  k_q+1 with p < q correspond to treks T_1 and T_2 in  which intersect. There are four cases depending on where this intersection occurs which are depicted in <Ref>. In the first three cases, there is a trek from k_p to k_q+1 through the intersection vertex — corresponding to a trek from i to j through m in <Ref>. This allows us to shorten the path in  by omitting k_p+q, …, k_q. The resulting path in  still has all its vertices contained in K.
The fourth case <Ref> goes from i over m to l, back up to m and then to j, omitting k from the path in . The walk in  traced out by this is a zig-zag with a tail m → l whose endpoint l is still in K.

For every pair of treks, multiple of these cases can occur simultaneously and each case can occur multiple times. One can freely choose any such intersection and resolve it according to <Ref> which results in a shortened path in . Iterating this procedure produces a simple path in  from i to j whose inner vertices are all contained in K. The union of the corresponding treks in  is a zig-zag with tails ⊆ as in <Ref>. 

By construction the subgraph  has the following property: for each collider m on , there is tail starting at m and completely contained in  whose end vertex is in K. Hence, <Ref> applies and shows that i j | K' on _ where K' is the set of vertices on  which are in K. Since  is a subgraph of , <Ref> implies that i j | K on _.








§ ACKNOWLEDGEMENTS


Tobias Boege was partially supported by the Academy of Finland grant number 323416 and by the Wallenberg Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. Benjamin Hollering was supported by the Alexander von Humboldt Foundation. Mathias Drton, Pratik Misra, and Daniela Schkoda acknowledge support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818). Sarah Lumpp acknowledges support by the Munich Data Science Institute (MDSI) at Technical University of Munich as part of the focus topic CAUSE. Daniela Schkoda and Sarah Lumpp were also supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research.


Part of this research was performed while some of the authors were visiting the Institute for Mathematical and Statistical Innovation (IMSI), which is supported by the National Science Foundation (Grant No. DMS-1929348). Benjamin Hollering and Tobias Boege also thank the Combinatorial Coworkspace (funded by Combinatorial Synergies SPP 2458). 

We thank Shiva Darshan for helpful discussions regarding the proof of <Ref>.














    


































plain







§ PROOFS




 §.§ Proof of <ref>


In this subsection we provide a proof of <ref> by constructing a specific submodel of the Lyapunov model and then inductively showing that the determinant of the submatrix Σ_iK, jK does not vanish for a trek  with endpoints i and j where the conditioning set K = V() ∖{i, j}. 

Let  be a trek with endpoints i and j such that the length of the left branch, whose sink is i, is ℓ and the length of the right branch, whose sink is j, is r. Let K = V() ∖{i, j} be the set of all other nodes on . Let '_⊆_ of the Lyapunov model on  obtained from the trek rule in <ref> by setting every edge parameter m_qp = 1 for all p→ q ∈ E(). In other words

    '_ = {Σ∈_n  : σ_ij = ∑_T : (ℓ, r)-trek
    from i to j 2 ζ^ℓ+r+1ℓ+rℓ}.

This follows directly from <ref> since every trek monomial m_T = ∏_e ∈ Tm_e = 1. Thus we can think of any matrix Σ∈'_ as a matrix whose entries are univariate polynomials in ζ. Our goal is now to show that for almost all Σ∈'_, |Σ_iK, jK| ≠ 0. To do this, we will focus on computing the lowest degree term in |Σ_iK, jK| as a polynomial in ζ.
Formally, the lowest degree term of a univariate polynomial f = ∑_k=k_0^n f_k ζ^k with f_k_0≠ 0 is (f)  f_k_0ζ^k_0.

Before we begin, note that every entry of every Σ∈_' is divisible by 2ζ. In the following sequence of lemmas, we will instead consider a slight alteration of the above parameterization given by

    ”_  = {Σ∈_n  : σ_ij = ∑_T : (ℓ, r)-trek
    from i to jζ^ℓ+rℓ+rℓ}.

This simply corresponds to factoring out the common 2ζ from each entry of Σ which does not change whether or not a minor of Σ vanishes. Thus if Σ exhibits a conditional dependence then 2ζ Σ will exhibit the same conditional dependence. To prove that our original model _ satisfies i j | K, it suffices to prove it for a matrix in _”.

The following easy facts are used throughout the proofs in this section:


  * The lowest degree term is multiplicative: (fg) = (f) (g).

  * Since  is a trek, there is a unique shortest trek between any two vertices and thus there is a well-defined notion of distance on the trek.

  * Let the shortest trek between i and j have length (ℓ, r). Since it is unique, we have (σ_ij) = ℓ+rℓζ^ℓ+r for every Σ = Σ(ζ) ∈_”.

  * An important special case of this formula occurs when i and j are on the same branch of . In that case ℓ = 0 or r = 0 and the binomial coefficient is 1.

We begin with the following lemma concerning the lowest degree terms of principal minors. 


































Let  be a trek and Σ∈_”. For any K ⊆ V we have (|Σ_K|) = 1. 



Under the parametrization of _”, any diagonal entry σ_ii has (σ_ii) = 1 coming from the empty trek on i, while every off diagonal entry has ((σ_ij)) ≥ 1. The result then follows immediately from the Leibniz formula for the determinant. 


In the following, we compute minors of the form |Σ_iK,jK| which are decisive for conditional independence by <Ref>. If K = V ∖{i,j}, then this minor is, up to sign, the adjoint entry (Σ)_ij. To facilitate inductive proofs via Schur's formula (which computes |Σ_iK,jK| using entries of (Σ_K)), it is important to not only show that these minors are nonzero, but also to consistently compute the signs.

 
Throughout this section we assume the following numbering of the vertices V = {1,2,…,n} of the trek: the leaf on the left branch is labeled 1, its parent 2, and so on, until the top node is labeled ℓ+1; from there we descend down with ℓ+2 and so on until the leaf on the right branch is labeled n = ℓ+r+1. Any minor |Σ_I,J| is computed by ordering the rows and columns of the matrix with respect to this ordering.




Let  be a trek and Σ∈_”. Let i, j ∈ V() be distinct and (ℓ, r) be the length of the shortest trek between them. For any K ⊆ V() ∖{i,j}, either |Σ_iK,jK| = 0 or ((|Σ_iK,jK|)) ≥ℓ+r. If i and j are adjacent, then (|Σ_iK,jK|) = ζ.



Consider the Leibniz formula

    |Σ_iK,jK| = ∑_π: iK → jK 
    bijective(π) ∏_k ∈ iKσ_k π(k).

Note that Σ_iK,jK is a matrix with a particular ordering on its rows and columns. This ordering fixes a bijection between iK and jK with respect to which (π) is well-defined.

It suffices to show that each summand is either zero or has lowest term of degree≥ℓ+r. Fix a bijection π: iK → jK. The image π(i) either equals j or is an element k_1 ∈ K. In the latter case, we may apply π again to get π(k_1) = π^2(i) which either equals j or k_2 ∈ K. Continuing in this way, collect all iterated images of i into a set A ⊆ iK including i and excluding the j which is eventually reached. Observe that

    (∏_k ∈ Aσ_k π(k))
      = ∑_k ∈ A(σ_k π(k)) ≥(σ_ij) = ℓ + r,

since the sequence (σ_i π(i)), (σ_π(i)π^2(i)), …, (σ_π^m(i)j) given by A encodes an undirected path from i to j which is at least as long as the shortest trek.

The lower bound in (<ref>) can only be attained with equality if A is completely contained in the shortest trek ' connecting i and j and if the successive images π(i), π^2(i), … are ordered such that π(i) is the closest node to i, π^2(i) the second closest and so on; while all elements of K ∖ A are fixed points of π and do not contribute to the degree. In this case, the σ_kk, k ∈ K ∖ A, have lowest term 1 and the shortest treks between adjacent members of the sequence i, π(i), π^2(i), …, j give a disjoint decomposition of ' hence achieving degree exactly ℓ+r in total. If i and j are adjacent, then ℓ+r = 1 and the only bijection π which satisfies this sends i to j directly and fixes every other k ∈ K. Since i and j are adjacent, they appear in the same place in the ordering of iK and jK, respectively, hence π corresponds to the identity permutation of the rows and columns and thus we compute (|Σ_iK,jK|) = (σ_ij) = ζ.


As the previous proof shows, the lower degree bound ℓ+r is achievable and the Leibniz formula always has summands of this degree. Whether or not these summands cancel to cause a bump in lowest degree depends on the locations of i and j on the trek and to each other, as well as the conditioning set K.

 
In the following proofs, we will use the Schur complement expansion of the determinant:

    _K(i,j) |Σ_iK,jK| = σ_ij |Σ_K| - Σ_i,K(Σ_K) Σ_K,j = σ_ij |Σ_K| - ∑_p,q ∈ Kσ_ip_pqσ_qj,

where _pq = (Σ_K)_pq. To use Schur's formula to expand the determinant by the (i,j)-entry of Σ_iK,jK as indicated, we have to permute the ith row as well as the jth column to be the first row and column, respectively. The symbol _K(i,j) denotes the sign change in the determinant incurred by this. This sign depends only on the difference of the positions of i in iK and j in jK. We will assume below that K contains all interior nodes on the shortest trek ' between i and j. In this case and if i ≠ j, _K(i,j) can be computed as

    _K(i,j) = (-1)^n' = (-1)^|j-i|+1,

where n' is the number of nodes on ', which equals |j-i|+1 under <Ref>.




Let  be a trek and Σ∈_”. Let i, j ∈ V() be distinct, non-adjacent vertices on the same branch of . Denote the length of the shortest path ' between i and j by ℓ > 1. If K ⊆ V() ∖{i,j} contains all interior nodes on ', then either
|Σ_iK,jK| = 0 or (|Σ_iK,jK|) > ℓ.





Without loss of generality, we may suppose by symmetry that both i and j are on the left branch of the trek and i < j. Then the shortest path connecting them is also the unique shortest trek connecting them, which has length (ℓ, 0).

We perform induction on ℓ≥ 2. The case ℓ = 2 can be handled using the methods of <Ref>. The shortest trek ' between i and j is the form i → k → j where k ∈ K by assumption. Using the terminology of the proof of <Ref>, the only two bijections iK → jK which contribute to the ζ^2 term are:


  * π_1 sending i ↦ j and fixing every other element. Since k is between i and j in the ordering, the element i in iK occupies the same place as k in jK; and k in iK the same place as j in jK. Hence π_1 corresponds to an adjacent transposition of the rows and columns and has (π_1) = -1. This summand is thus -ζ^2.

  * π_2 sending i ↦ k ↦ j and fixing every other element. By similar reasoning as in the previous case, this corresponds to the identity permutation of the rows and columns and thus the contribution is ζ^2.

The two summands cancel and therefore either the entire determinant vanishes or the lowest term has degree > 2.

For the induction step, we use the Schur complement expansion (<ref>). By <ref> and the definition of _”, we know (σ_ij|Σ_K|) = ζ^ℓ. The goal is to show that for each choice of p, q ∈ K, either σ_ip_pqσ_qj = 0 or (σ_ip_pqσ_qj) ≥ℓ and in case of equality to compute the coefficient of the ζ^ℓ term.

We distinguish four cases (up to exchanging p and q or switching branches) depending on the locations of p and q with respect to i and j:


  * In case of <Ref>, notice that the shortest trek from i to p strictly includes the entire path from i to j, so ((σ_ip_pqσ_qj)) ≥((σ_ip)) > ℓ.

  * In case of <Ref>, the reason is similar. <Ref> shows ((σ_ip_pq)) > ℓ.

  * In case of <Ref>, we know by <Ref> that ((_pq)) > ℓ.

  *  The last case is depicted in <Ref>.

Then (σ_ip) = ζ^a and (σ_qj) = ζ^b and a+b+c=ℓ. So

    ((σ_ip_pqσ_qj)) = ℓ((_pq)) = c.

Notice that p and q are on the same branch of . If they coincide (c=0), then (_pq) = 1 by <Ref>; if they are adjacent (c=1), <Ref> yields _pq = -|Σ_K∖{q},K∖{p}| = -ζ. Otherwise, the induction hypothesis ensures _pq = 0 or (_pq) > c, which do not contribute to the lowest term. 

This implies that

    (∑_p,qσ_ip_pqσ_qj)    = ∑_p, q 
     c = 0(ζ^a · 1 ·ζ^b) + ∑_p, q 
     c = 1(ζ^a · (-ζ) ·ζ^b) 
       = ℓζ^ℓ - (ℓ - 1)ζ^ℓ = ζ^ℓ.


Thus the only case which contributes to the ζ^ℓ term in |Σ_iK,jK| is <ref>. Its contribution cancels with (σ_ij|Σ_K|) = ζ^ℓ in (<ref>), implying the result. 




Let  be a trek and i, j ∈ V() be nodes on different branches of  (in particular neither of them is the top node). Denote the length of the shortest path ' between i and j by (ℓ, r). If K ⊆ V() ∖{i,j} contains all interior nodes on ', then there exists Σ∈_” such that i j | K. 



Without loss of generality, we may assume that i < j in the ordering, so that i is on the left branch and j on the right. We claim that (|Σ_iK,jK|) = (-1)^ℓ+r+1ℓ+r-2ℓ-1ζ^ℓ+r which shows that |Σ_iK,jK| is non-zero and yields conditional dependence for all but a finite number of values of ζ.

We prove this statement by induction on ℓ, r ≥ 1, again using the Schur complement expansion (<ref>),

    (-1)^ℓ+r+1 |Σ_iK, jK| = σ_ij|Σ_K|  - Σ_i,K(Σ_K) Σ_K,j = σ_ij|Σ_K| - ∑_p,q ∈ Kσ_ip_pqσ_qj

with _pq = (Σ_K)_pq.

By <ref> and the definition of _”, we have (σ_ij |Σ_K|) = ℓ+rℓζ^ℓ+r.

The other part of the expansion is a sum over σ_ip_pqσ_qj which we deal with term by term, so fix p, q ∈ K. If p is strictly closer to j than q is (or, symmetrically, q is closer to i than p is), then (σ_ipσ_qj) > (σ_ipσ_pj) ≥(σ_ij) = ℓ+r. Additionally, by <Ref>, p and q must be contained in V('), i.e. i < p ≤ q < j in our ordering of vertices by <Ref>.

Hence, to find the lowest-order terms, we only need to consider

    p, q ∈ V(') ∖{i,j}⊆ K such that i < p ≤ q < j.


With these remarks, the basis of induction is easy to show. For ℓ = r = 1 the only possible option in (<ref>) is that p = q = top node of. Then (σ_ip_pqσ_qj) = ζ· 1 ·ζ = ζ^2. With (σ_ij |Σ_K|) = 2 ζ^2 we get (|Σ_iK,jK|) = (-1)^3 · (2ζ^2 - ζ^2) = -ζ^2, as required.



In the general case, the reduction via (<ref>) allows us to identify all relevant choices of p,q with decompositions of ' into three disjoint parts: the trek i to p, the trek p to q and the trek q to j.

There are three cases:



  * Suppose that p and q are both on the left branch of ' and suppose that they are not both equal to the top node. Then the trek decomposes into four paths as in <Ref>. We have (σ_ipσ_qj) = c+rcζ^a+c+d. By <Ref>, the lowest exponent of σ̂_pq in this case is b if and only if b = 0 or b = 1. Otherwise it is larger than b and a+b+c+d > ℓ+r disqualifying those cases from contributing to the lowest terms of the overall expression, so assume b = 0 or b = 1. Then (σ̂_pq) = (-1)^bζ^b by <Ref>. In total, this case contributes ∑_c = 1^ℓ-1c+rc when b=0 and -∑_c=0^ℓ-2c+rc when b=1 to the lowest-order terms. (The different summation ranges for c reflect that we must have a ≥ 1 and a+c = ℓ and also that if b = 0, in order for p = q not to be the top node, we must have c ≥ 1 whereas if b = 1, we only need c ≥ 0.) Together, this gives ℓ+r-1ℓ-1-1. The case where p and q are both on the right branch is analogous and contributes ℓ+r-1r-1 = ℓ+r-1ℓ-1.


  * Suppose that p is on the left branch and q is on the right branch but neither of them is the top node. Then the trek decomposes into four paths as in <Ref> and (σ_ipσ_qj) = ζ^a+d. Here the induction hypothesis applies and computes |Σ_pL,qL| = (-1)^q-p+1b+c-2b-1ζ^b+c, where L = K ∖{p,q}. (Note that L contains all interior nodes on the shortest trek between p and q since K had the same property for i and j.) However, there is another sign to account for coming from the cofactor: _pq = (Σ_K)_pq = (-1)^q-p |Σ_pL,qL| = -b+c-2b-1ζ^b+c. The sign only depends on q-p, again because L contains all vertices between them in the ordering.

In total, this case contributes -∑_b=1^ℓ-1∑_c=1^r-1b+c-2b-1 = 1 - ℓ+r-2ℓ-1 to the lowest term.


  * There is a special case when p and q both coincide with the top node of the trek, as in <Ref>. In this case, they are both simultaneously on the left and the right branch but this case was not counted in the previous bullet points. As in the base case of the induction, the contribution is (σ_ip_pqσ_qj) = ζ^ℓ+r.

In total,

    (-1)^ℓ+r+1(|Σ_iK,jK|) 
       = ( ℓ+rℓ - ℓ+r-1ℓ-1 - ℓ+r-1ℓ_= 0 + 2 
     - (1 - ℓ+r-2ℓ-1) - 1 ) ζ^ℓ+r
       = ℓ+r-2ℓ-1ζ^ℓ+r,

as claimed.


The last remaining case needed to complete the proof of <ref> is the case when  is a trek such that ℓ = 0 or r = 0, meaning that  is actually a path. In this case <ref> shows that the lowest degree term can have degree strictly higher than ℓ + r. For this reason, we instead analyze the highest degree term in ζ which appears in the expansion of |Σ_iK, jK|. Let (f) denote the highest degree non-zero term of a univariate polynomial f.



Let  be a directed path from i to j and K=V()∖{i,j} be the remaining vertices. Then there exists Σ∈_” such that i j | K. 


To show that there exist Σ∈_” for which i j | K, it is enough to show that |Σ_iK,jK| as a univariate polynomial in ζ has a non-zero coefficient.

To this end, we compute the determinant of (Σ_iK,jK), which is the matrix obtained by taking the highest degree term of each entry of Σ_iK,jK, at ζ=1. If this value is non-zero, it follows that |(Σ_iK,jK)| ≠ 0 is the highest degree term of |Σ_iK,jK| which is thus non-zero as well.

We claim that |(Σ_iK,jK)|=1 at ζ=1.
Recall that _” is defined by the parametrization which maps σ_ij to the sum of ζ^ℓ+rℓ+rℓ over all possible treks between i and j. Hence, the entries of (Σ_iK,jK) only take into account the longest treks between nodes (with the highest value of ℓ+r).

For simplicity, we number the vertices from 1 to n with i=1 and j=n. Now, the longest and unique trek between 1 and k for any k≤ n is precisely the directed path from 1 to k. However, the longest trek between a and k for any 1<a<k is one where the left and right branches are directed paths from 1 to a and 1 to k, respectively. This implies that the term with highest degree for σ_1k has coefficient 1 (as ℓ=0) and for that of σ_ak has coefficient a+k-2a-1. This gives us that the (a,b) entry of (Σ_{1,2,3,…, n-1},{2,3,…,n}) at ζ=1 is precisely a+b-2a-1. However, using the identity in <cit.>, we can obtain an LU decomposition of this matrix where the diagonal terms of both L and U are 1. This implies that |(Σ_iK,jK)|=1 at ζ=1, which proves our claim. 




 §.§ Proof Details for <ref>




    In the proof of <ref>, we solve the following six equations resulting from the block-matrix partition for the corresponding blocks of Σ^(m) for m →∞:
    
        
  (A) M^*_UΣ^(m)_U + Σ^(m)_U[-0]M^*_U + 2_k-1 = 0 

        This equation is uniquely solved by Σ^(m)_U = Σ^*_U due to <ref>.
        
        
  (B) M^*_UΣ^(m)_U,k + mΣ^(m)_U,k-1 -m Σ^(m)_U,k = 0 

        The matrix M^*_U - m_k-1 is invertible due to the structure of M^*, so <ref> together with (A) yields
        
    lim_m →∞Σ^(m)_U,k = lim_m →∞Σ^(m)_U,k-1 = Σ^*_U,k-1.

        
        
  (C) M^*_UΣ^(m)_U,W + Σ^(m)_U,WM^*_ψ(W) + Σ^(m)_U,k(m^*_k,k-1, 0, …, 0) = 0 

        Taking the limit of the left-hand side for m →∞ yields
        
    M^*_UΣ^(m)_U,W + Σ^(m)_U,WM^*_ψ(W) + Σ^(m)_U,k (m^*_k,k-1, 0, …,0)  
        M^*_U( lim_m →∞Σ^(m)_U,W) + ( lim_m →∞Σ^(m)_U,W) M^*_ψ(W) + Σ^*_U,k-1 (m^*_k,k-1, 0, …,0),

        where we employed (B) in the last summand. This last summand can further be rewritten as 
        
    Σ^*_U,k-1 (m^*_k,k-1, 0,…,0) = Σ^*_UM^*_ψ(W),U

        by attaching a (k-2) × (k-1) zero matrix to the existing row vector and thereby extending it to M^*_ψ(W),U. Setting the limit to zero yields equation <ref> which is solved by Σ^*_U,ψ(W). This gives us lim_m →∞Σ^(m)_U,W = Σ^*_U,ψ(W).
        
        
  (D) 2-2mΣ^(m)_k,k+m(Σ^(m)_k,k-1+Σ^(m)_k-1,k) = 0 

        Solving for Σ^(m)_k,k, letting m go to infinity, and using the result from (B) yields 
        
    lim_m →∞Σ^(m)_k,k 
                = lim_m →∞(1/m + Σ^(m)_k,k-1+Σ^(m)_k-1,k/2) 
                = Σ^*_k-1,k-1+Σ^*_k-1,k-1/2 
                = Σ^*_k-1,k-1.

        
        
  (E) m Σ^(m)_k-1,W - m Σ^(m)_k,W+ Σ^(m)_k,k (m^*_k,k-1, 0,…,0) + Σ^(m)_k,WM^*_ψ(W) = 0 

        The matrix M^*_ψ(W) - m_-k is invertible due to the structure of M^*, so the transposed version of <ref> together with (C) and (D) yields
        
    lim_m →∞Σ^(m)_k,W = lim_m →∞Σ^(m)_k-1,W = Σ^*_k-1,ψ(W).

        
        
  (F) M^*_ψ(W)Σ^(m)_W + Σ^(m)_WM^*_ψ(W) + 2_-k

        + (m^*_k,k-1, 0,…,0)^T Σ^(m)_k,W + Σ^(m)_W,k(m^*_k,k-1,0,…,0) = 0 

        We use that Σ^(m)_k,W→Σ^*_k-1,ψ(W) as well as Σ^(m)_W,k→Σ^*_ψ(W),k-1 for m →∞ due to (E) and take the limit on both sides. This yields for the left-hand side
        
    M^*_ψ(W)Σ^(m)_W + Σ^(m)_WM^*_ψ(W) + 2_-k
            + (m^*_k,k-1, 0,…,0)^T Σ^(m)_k,W + Σ^(m)_W,k(m^*_k,k-1,0,…,0) 
        M^*_ψ(W)(lim_m →∞Σ^(m)_W) + (lim_m →∞Σ^(m)_W) M^*_ψ(W) + 2_-k
            + (m^*_k,k-1, 0,…,0)^T Σ^*_k-1,ψ(W) + Σ^*_ψ(W),k-1(m^*_k,k-1,0,…,0) 
    
                =      M^*_ψ(W)(lim_m →∞Σ^(m)_W) + (lim_m →∞Σ^(m)_W) M^*_ψ(W) + 2_-k
            + M^*_ψ(W),UΣ^*_U,ψ(W) + Σ^*_ψ(W),UM^*_ψ(W),U,

        where we employed the same technique of rewriting the last two summands as for (C). Setting the limit to zero yields equation <ref> that is solved by Σ^*_ψ(W). We conclude lim_m →∞Σ^(m)_W = Σ^*_ψ(W).
     


 
    Let A^(m) and C^(m) be convergent sequences of × l matrices, and let D,E,F ∈^× be constant matrices such that E - m _ is invertible for all m ∈_> 0.
    If a further matrix sequence B^(m) satisfies that 
    
    m · A^(m) - m · B^(m) + D · A^(m) + E · B^(m) + F · C^(m) =0

    for all m ∈_> 0, then
    
    lim_m→∞ B^(m) = lim_m→∞ A^(m).




    By collecting the terms and dividing by m, we have 
    
    (1/m D + _) A^(m) + (1/m E - _) B^(m) +  1/m F · C^(m) = 0.

    Since (1/m E - _) is invertible, we can write
    
    B^(m) = - (1/m E - _)^-1((1/m D + _) A^(m) - 1/m F · C^(m)).

    As matrix inversion is continuous and A^(m) and C^(m) are convergent, letting m go to infinity on both sides yields the result.





 Paper Title: Outliers and bounded rank perturbation for non-Hermitian random band matrices.





§ ABSTRACT 


In this work we consider general non-Hermitian square random matrices X 
that include a wide class of random band matrices with independent entries. Whereas the existence of limiting density is largely unknown for these inhomogeneous models, we show that spectral outliers can be determined under very general conditions when perturbed by a finite rank deterministic matrix. More precisely, we show that whenever 𝔼[X]=0,𝔼[XX^*]=𝔼[X^*X]=1 and 𝔼[X^2]=ρ1, and under mild conditions on sparsity and entry moments of X, then with high possibility all eigenvalues of X are confined in a neighborhood of the support of the elliptic law with parameter ρ. Also, a finite rank perturbation property holds: when X is perturbed by another deterministic matrix C_N with bounded rank, then the perturbation induces outlying eigenvalues whose limit depends only on outlying eigenvalues of C_N and ρ. This extends the result of Tao <cit.> on i.i.d. random matrices and O’rourke and Renfrew on elliptic matrices <cit.> to a family of highly sparse and inhomogeneous random matrices, including all Gaussian band matrices on regular graphs with degree at least (log N)^3. A quantitative convergence rate is also derived. We also consider a class of finite rank deformations of products of at least two independent elliptic random matrices, and show it behaves just as product i.i.d. matrices.





Bound states in doped charge transfer insulator
    Tao Xiang
    August 5, 2024
===============================================






§ INTRODUCTION


In this work we consider the spectral properties of a very general class of inhomogeneous non-Hermitian random matrices, including a wide class of random band matrices. 


To put our results in a proper context, we first recall what is known for a random matrix with i.i.d. entries, which has a fully homogeneous variance profile.


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.
 


  * (The circular law, see <cit.> and references therein)  The empirical spectral density of n^-1/2A_n, defined as n^-1∑_j=1^nδ_λ_j with (λ_j)_1≤ j≤ n the eigenvalues of n^-1/2A_n, converges to the circular law, i.e. the uniform distribution in the unit disk in the complex plane. 

  * (No outliers, see <cit.> <cit.>) Let ρ(n^-1/2A_n) denote the spectral radius of n^-1/2A_n. Assuming the atom distribution a_ij has a finite fourth moment: 𝔼[|a_ij|^4]<∞, then ρ(n^-1/2A_n) converges to 1 almost surely.


  * (Low rank perturbation, see <cit.>) Assuming the atom distribution a_ij has a finite fourth moment. Fix some ϵ>0 and let C_n be deterministic matrix with O(1) rank and operator norm for each n, having no eigenvalues with norm in [1+ϵ,1+3ϵ] and j eigenvalues λ_1(C_n),⋯,λ_j(C_n) for a j=O(1) in {z∈ℂ:|z|>1+3ϵ}. Then almost surely when n is sufficiently large, there are exactly j eigenvalues λ_1(n^-1/2A_n+C_n),⋯,λ_j(n^-1/2A_n+C_n) of n^-1/2A_n+C_n in {z∈ℂ:|z|>1+2ϵ} and after labeling, λ_i(n^-1/2A_n+C_n)=λ_i(C_n)+o(1) for each 1≤ i≤ j.





    The finite fourth moment condition 𝔼[|a_ij|^4]<∞ is not necessary for the no outliers result in (2). We only need a finite variance, see <cit.>, <cit.>. 


    

 The corresponding problem in (3) low rank perturbation  for Hermitian random matrices has been studied in more detail in <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and see also references therein. There is a phase transition called the BBP transform: Consider a Wigner matrix W_n, a unit vector v and θ>0, then 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. Item (3) of Theorem <ref> shows that when we consider instead a square matrix with i.i.d. entries, then eigenvalues outside the unit circle converge to eigenvalues of the perturbation. 
 
 



Inhomogeneous random matrix models have attracted much recent interest, from both the theoretical and applied perspective. Random band matrix is one of the most important inhomogeneous matrix models in modern mathematical physics <cit.>. In this paper we investigate a non-Hermitian version of random band matrix, which can be defined in the following general form.

(Non-Hermitian random matrix on regular graphs)
Let N be an integer and d_N∈ [1,N] be an integer depending on N. 

Fix any d_N-regular directed graph G_N=([N],E_N) having N vertices and a set E_N of directed edges on [N]. In this graph, self loops are allowed but multiple edges are excluded.
We further require that either one of the following two alternatives hold

    
  * There is a self loop (x,x)∈ E_N for all x∈ [N],
    
  * There is no self loop (x,x)∈ E_N for any x∈ [N].


Consider a family of mean 0, variance 1 (real or complex) random variables g_xy for each (x,y)∈ E_N. These random variables are assumed to be independent but not necessarily of the same distribution. However for the diagonal entries we require 𝔼[g_xx^2]=D for each (x,x)∈ E_N for some fixed constant D with |D|≤ 1.


Then we define an N× N random matrix X_N by setting (X_N)_xy=1_(x,y)∈ E_Ng_xy.





 This definition is sufficiently general as it encompasses a wide class of distinct random matrix models. They include periodically banded matrix where we set (x,y)∈ E_N if min(|x-y|,N-|x-y|)≤ d_N for a bandwidth d_N. They also include block matrix which is the direct sum of N/d_N blocks of i.i.d. matrix of size d_N. They further include the direct sum of block and band matrices, and many other possibilities.  


Now we state the main result of this paper, applied to the matrix (d_N)^-1/2X_N. 

  
Assume that the regular digraphs G_N are increasing (E_N⊂ E_N+1 for each N) and that we can embed X_N as the top-left N× N submatrix of X_N+1, so that all the random matrices X_N are defined on the same probability space.

     Assume moreover that
    one of the following three conditions hold:
    
        
  * g_xy are standard (real or complex) Gaussian variables and d_N>>(log N)^3, or
        
  * g_xy are real or complex random variables that satisfy
        
    lim_N→∞ (log N)^2 max_(x,y)∈ E|(d_N)^-1/2g_xy|=0   a.s.,

        
  * g_xy are i.i.d. real centered variance 1 random variables such that, for some p>4, we have  𝔼[|g_xy|^p]<∞ and d_N>>N^2/p-2(log N)^4p/p-2.
     
    
    Then the random matrix (d_N)^-1/2X_N in Definition <ref> satisfies the following properties:
    
        
  * For any ϵ>0, almost surely as N is sufficiently large, ρ((d_N)^-1/2X_N)≤ 1+ϵ.
    
    
  * Moreover, fix some ϵ>0, let C_N be deterministic matrix with O(1) rank and operator norm for each N, having no eigenvalues with norm in [1+ϵ,1+3ϵ] and j eigenvalues λ_1(C_N),⋯,λ_j(C_N) for a j=O(1) in {z∈ℂ:|z|>1+3ϵ}.
    
    
    Then almost surely as N is large, there are exactly j eigenvalues λ_1((d_N)^-1/2X_N+C_N),⋯,λ_j((d_N)^-1/2X_N+C_N) of (d_N)^-1/2X_N+C_N in {z∈ℂ:|z|>1+2ϵ} and after labeling, 
    λ_i((d_N)^-1/2X_N+C_N)=λ_i(C_N)+o((loglog N)^-1/2)
 for each 1≤ i≤ j.
    


    
     
  


     A discussion on the quantitative estimates in this theorem are presented in Remark <ref>. 









 §.§ The elliptic case


In this section we state an elliptic analogue of Theorem <ref>.

Recall that an N× N complex random matrix E_N=(e_ij)_i,j=1,⋯,N is said to be elliptic if each entry e_ij has mean zero and variance one, that the covariance Cov(e_ij,e_ji)=ρ∈(-1,1) for each i≠ j, and that all entries of E_N are independent modulo symmetry constraint. This definition interpolates the i.i.d. case ρ=0 and the symmetric case ρ=1.
Under some additional assumptions, it is proved <cit.> that the empirical eigenvalue density of N^-1/2E_N converges to the elliptic law, that is, the uniform distribution on the ellipse ℰ_ρ 

    ℰ_ρ:={z∈ℂ:Re(z)^2/(1+ρ)^2+Im(z)^2/(1-ρ)^2≤ 1}

in the complex plane. 
Assuming further that e_ij has a finite fourth moment, <cit.> proved that a finite rank perturbation result also holds for random elliptic matrices, which in the i.i.d. case ρ=0 reduces to Theorem <ref> (3) (i.e., the result of <cit.>) and in the Wigner case ρ=1 reduces to the result of <cit.>
when the perturbation matrix C_n is self-adjoint.

Analogously to Theorem <ref>, we now consider random band matrices with elliptic correlations, which generalizes Hermitian random band matrices and random band matrices with independent entries in Definition <ref>. As will be explained in Section <ref>, it is presently unknown how to prove the limiting eigenvalue density exists and converges to the elliptic law for a random band matrix with elliptic correlation, at least when the bandwidth b_N≤ N^33/34. However, we show in the following theorem that outliers in the banded model can still be determined under elliptic correlation, thus generalizing <cit.> to a wide class of band matrices with elliptic correlation.

In contrast to previous work, we can consider any covariance ρ∈ℂ:|ρ|≤ 1 without assuming ρ is a real number. In this case we need to define the region ℰ_ρ slightly differently as follows, for any ρ∈ℂ, |ρ|≤ 1,

    ℰ_ρ:={z∈ℂ:z=x+ρx̅,   for some |x|≤ 1,   x∈ℂ}.

We prove that the new definition of ℰ_ρ degenerates to the old one when ρ is real:

    Assume that ρ∈ [-1,1]ℝ, then the set ℰ_ρ defined in (<ref>) coincides with the ellipse defined in (<ref>). In general, we have 
    
    ℰ_ρ=e^i(ρ)/2ℰ_|ρ|,
 that is, ℰ_ρ is ℰ_|ρ| rotated by the angle (ρ)/2 in the counterclockwise direction across the origin.
 
This lemma follows from routine computations, and see Section <ref> for the proof. 

For any ϵ>0 we denote by ℰ_ρ,ϵ the following region which is a neighborhood of ℰ_ρ in the complex plane: 

    ℰ_ρ,ϵ:={z∈ℂ:dist(z,ℰ_ρ)≤ϵ}.
 We also use the notation 𝔻 to mean the unit disk in the complex plane 𝔻:={z∈ℂ:|z|≤ 1}, and for any ϵ>0, (1+ϵ)𝔻 denotes the set {z∈ℂ:|z|≤ 1+ϵ}. Clearly, ℰ_0,ϵ=(1+ϵ)𝔻.


We now state the main result concerning elliptic band matrices:
 Let N∈ℕ_+ and d_N∈[1,N]. Consider a family of undirected d_N-regular graphs G_N=([N],E_N) such that either all self loops (x,x),x∈[N] do not exist in E_N, or all such self loops (x,x),x∈[N] are edges in E_N with multiplicity one.  


Fix a covariance parameter ρ∈ℂ:|ρ|≤ 1.
Define a random matrix X_N on [N]^2 such that for any x<y, (x,y)∈ E_N, we set ((X_N)_(x,y),(X_N)_(y,x)) to be a pair of mean 0, variance 1 random variables such that 
    𝔼[(X_N)_(x,y)(X_N)_(y,x)]=ρ.

If (x,y)∉ E_N we set the pair to be (0,0). For self loops (x,x)∈ E we set (X_N)_(x,x) an i.i.d copy of a mean 0, variance 1 random variable g_0. We assume that all the random variables ((X_N)_(x,y),(X_N)_(y,x))_(x,y)∈ E_N are mutually independent over the edges (x,y)∈ E_N. 

We assume that the graphs G_N are increasing (so that E_N⊂ E_N+1) and that X_N can be embedded as the [1,N]× [1,N] submatrix of X_N+1, so that all the {X_N} can be realized on the same probability space. 

For the entry distributions, we further assume that either one of the following three conditions hold: 
    
  * Either d_N>>(log N)^3 and the matrix X_N has jointly Gaussian entries;  
  * Or d_N>>(log N)^5 and almost surely we have |(X_N)_(x,y)|≤ C for some fixed C>0 and all edges (x,y)∈ E_N; or 
  * For each (x,y)∈ E_N:x<y, the pair ((X_N)_(x,y),(X_N)_(y,x)) has the same law as (g_1,g_2), where g_1,g_2 are two real-valued symmetric random variables of mean 0, variance 1, having the same distribution: g_1law=g_2 and 𝔼[g_1g_2]=ρ. We require (g_0,g_1,g_2) satisfy that, for some p>4 and d_N>>N^2/p-2(log N)^4p/p-2, we have 𝔼[|g_i|^p]<∞,i=0,1,2.

Then we have the following conclusion: 
    
  * For any ϵ>0, almost surely as N tends to infinity there is no eigenvalue of (d_N)^-1/2X_N in ℂ∖{ℰ_ρ,ϵ.


  * Moreover, fix ϵ>0 and consider C_N a family of fixed N by N matrix with O(1) rank and operator norm, having no eigenvalues that satisfy 

    λ_i(C_N)+ρ/λ_i(C_N)∈ℰ_ρ,3ϵ∖ℰ_ρ,ϵ and  |λ_i(C_N)|>1,

and having j=O(1) eigenvalues λ_1(C_N),⋯,λ_j(C_N) such that 
    λ_j(C_N)+ρ/λ_j(C_N)∈ℂ∖ℰ_ρ,ϵ, and |λ_j(C_N)|≥ 1.
 

Then almost surely for N large enough, we can find j eigenvalues of (d_N)^-1/2X_N+C_N: λ_i((d_N)^-1/2X_N+C_N), i=1,⋯,j in ℂ∖ℰ_ρ,2ϵ that satisfy, after proper relabeling, 
    λ_i((d_N)^-1/2X_N+C_N)=λ_i(C_N)+ρ/λ_i(C_N)+o((loglog N)^-1/2)
 for each 1≤ i≤ j.










 §.§ A master theorem


In this paper we will not prove Theorem <ref> and <ref> directly, but rather deduce them as special cases of the following master theorem. 

A major observation here is that the regular graph structure in Theorem <ref> and <ref> are not the most intrinsic condition: for an N by N random matrix X with 𝔼[X]=0, the intrinsic condition guaranteeing ellipticity should be 
𝔼[X^2]=ρ1 for some fixed constant ρ. The assumption that the variance profile is doubly stochastic, namely 𝔼[XX^*]=𝔼[X^*X]=1, is sufficiently general to cover both the homogeneous case of a standard i.i.d. and standard elliptic matrix, and the band matrix case considered in Theorem <ref> and <ref>. We will show this is indeed the case: for a very general random matrix X with mean zero and satisfying some constraints, then its behavior in terms of outliers is completely determined whenever 𝔼[X^2]=ρ1 and 𝔼[XX^*]=𝔼[X^*X]=1 for some |ρ|≤ 1. This generalizes the Hermitian case where ρ=1, as considered in Theorem 2.7 of <cit.>.

Before presenting our master theorem, we introduce a few notations on very general random matrices from <cit.> and <cit.>, 


In the Gaussian case, consider the following random matrix model as in <cit.>, Section 1. Consider fixed matrices A_0,⋯.A_n∈ M_N(ℂ), where M_N(ℂ) denotes the set of N by N square matrices with complex entries. Let g_1,⋯,g_n be standard real Gaussian variables with mean 0 and variance 1. Consider a canonical model 
    X:=A_0+∑_i=1^n g_i A_i.
 It is clear that any square Gaussian random matrix admits a decomposition as (<ref>). We need to introduce a few more parameters on X.


For a Gaussian matrix X (<ref>) we define its covariance profile Cov(X)∈ M_N^2(ℂ) via 

    Cov(X)_ij,kl=∑_s=1^n (A_s)_ij(A_s)_kl
 and we now define 

    v(X)^2=Cov(X)=sup_Tr|M|^2≤ 1∑_s=1^n [Tr[A_sM]]^2,
       σ(X)^2:=𝔼[(X-𝔼X)^*(X-𝔼(X)]∧𝔼[(X-𝔼X)(X-𝔼(X)^*],
       σ_*(X)^2:=sup_x=w=1𝔼[|⟨ v,(X-𝔼X)w⟩|^2],

and finally define 
    ṽ(X)^2:=v(x)σ(X).


Our general theorem in the Gaussian case is as follows
 Fix ρ∈ℂ with |ρ|≤ 1.
    Consider X an N
    × N Gaussian random matrix with 
    𝔼[X]=0,  𝔼[XX^*]=𝔼[X^*X]=1,  𝔼[X^2]=(ρ+D_N)1,
 where D_N are fixed complex numbers such that lim_N→∞(log N)D_N=0.
    
    
    Assume further that 
    ṽ(X)≤ (log N)^-1,  σ_*(x)≤ (log N)^-1.5.
 
    
    
    Then for any fixed ϵ>0, we can 
 find C_1>0 a fixed constant depending only on ρ and ϵ, such that
    
        
  * With probability at least 1-C_1N^-1.5 there is no eigenvalue of X lying in ℂ∖ℰ_ρ,ϵ.

  * Consider deterministic N by N matrices C_N of rank O(1) and operator norm O(1), having no eigenvalues that satisfy 

    λ_i(C_N)+ρ/λ_i(C_N)∈ℰ_ρ,3ϵ∖ℰ_ρ,ϵ and  |λ_i(C_N)|>1,

and having j=O(1) eigenvalues λ_1(C_N),⋯,λ_j(C_N) such that 
    λ_j(C_N)+ρ/λ_j(C_N)∈ℂ∖ℰ_ρ,ϵ, and |λ_j(C_N)|≥ 1.
 

Then with probability at least 1-C_1N^-1.4, there exists precisely j eigenvalues λ_i(X+C_N), i=1,⋯,j of X+C_N in ℂ∖ℰ_ρ,2ϵ and, after a possible relabeling of the eigenvalues, we have 
    λ_i(X+C_N)=λ_i(C_N)+ρ/λ_i(C_N)+o((loglog N)^-1/2)
 for each 1≤ i≤ j,
 

 



We have presented our main theorem with quantified probability and convergence estimates. These estimates are not meant to be optimal and are of illustrative use only. The probability estimate 1-C_1N^-1.4 can easily be upgraded to be 1-C_1N^-p  for any p∈ℕ_+, and we use the latter as it is already sufficient for applying Borel-Cantelli lemma. 

The error term (loglog N)^-1/2 is more delicate and can be replaced by any quantity that vanishes strictly slower than (log N)^-p and any p>0. We choose the term (loglog N)^-1/2 for rotational simplicity only. This appears to be the first time where an explicit convergence rate is proven for elliptic random matrices.

An interesting problem is whether the error term can decay faster than (log N)^-p: we believe this cannot hold unconditionally but instead relies strongly on the structure of the random matrix X. For a square matrix with i.i.d. entries, fluctuations of outlying eigenvalues on the scale N^-1/2 around their deterministic limit are studied in <cit.>. As we consider far more general non-Hermitian random matrices, we do not expect a similar pattern of fluctuations should hold for typical band matrices considered in this paper.




In the general case, consider the following random matrix model as in <cit.>


    X:=Z_0+∑_i=1^n Z_i
 where Z_1,⋯,Z_n are independent N by N random matrices having mean zero. 


We can define v(X),σ(X) and σ_*(X) analogously as in the Gaussian model.
We also define the following function for X, where Z_i is the operator norm of Z_i:

    R(X):=max_1≤ i≤ nZ_i_∞,  R̅(X):=𝔼[max_1≤ i≤ nZ_i^2]^1/2.

Then our master theorem for non-Gaussian X is as follows:


    Let X be an N by N random matrix in the form (<ref>). Assume that  
    𝔼[X]=0,  𝔼[XX^*]=𝔼[X^*X]=1,  𝔼[X^2]=(ρ+D_N)1,
 where D_N are fixed complex numbers satisfying lim_N→∞(log N)D_N=0 . Assume that 
    ṽ(X)≤ (log N)^-1,  σ_*(x)≤ (log N)^-3/2,  R(X)<<(log N)^-2.
 Then all the conclusions in Theorem <ref> continue to hold for X.




    The constant D_N→ 0 in Theorem <ref> and <ref> is used to take care of the diagonal entry of X, as they lead to a different computation when we compute 𝔼[X^2].





 Theorem <ref> weakens a number of hypotheses that are traditionally imposed in the elliptic random matrix literature.
    In prior works <cit.>, <cit.> and <cit.>, they consider an elliptic matrix where each pair (g_ij,g_ji) is independently distributed as a pair (g_1,g_2), where (g_1,g_2) belongs to the so-called (μ,ρ) family (<cit.>, Definition 1.6). This definition of (μ,ρ) family requires the real and imaginary parts of g_1,g_2 are independent and the covariance of the real and imaginary parts of g_i should have a very specific form. Our theorem <ref> states that such restrictions are not necessary, and we do not even need (g_ij,g_ji) (g_kl,g_lk) are identically distributed: their real and imaginary parts can have a completely different covariance matrix for different edges (i,j),(k,l)∈ E, (but we do require diagonal entries to be i.i.d.). As another weakening, we can consider any covariance ρ∈ℂ,|ρ|≤ 1 without assuming ρ is real, as in <cit.>, <cit.> and <cit.>. In these cited works, the cases ρ=± 1 are excluded as the least singular value bound in <cit.> no longer holds: in the Wigner case ρ=1 this still holds with a separate proof <cit.> whereas in the anti-symmetric case ρ=-1 the matrix is always singular in odd dimensions. In our paper we do not rely on least singular value estimates as in <cit.>, so we treat |ρ|=1 in a unified way as for |ρ|<1, and in particular we cover the anti-symmetric case ρ=-1 which was largely unexplored before.

    



    Theorem <ref> and <ref> also cover (homogeneous) sparse i.i.d matrices and elliptic matrices. Indeed, we may take G to be the complete graph on [N] and take random variables g_xy to be g_xy=b_xyh_xy where h_xy is a Ber(k_N/N) random variable taking value 1 with probability k_N/N and 0 otherwise, and h_xy are bounded random variables having mean 0 and variance 1, independent from h_xy. Then clearly (N/k_N)^1/2g_xy has mean 0 and variance 1, and as long as k_N>(log N)^5 we have that (N/k_N)^1/2g_xy satisfy the assumptions of Theorem <ref>, (2) with g_xy replaced by (N/k_N)^1/2g_xy and d_N:=N. Indeed, the proof of Theorem <ref> (2) allows distributions of g_xy to be N-dependent so long as all quantitative estimates are satisfied. This establishes finite rank perturbations for k_N-sparse (homogeneous) i.i.d. matrices whenever k_N>>(log N)^5, which also appears to be new. 
    
    We note that due to the homogeneous structure and i.i.d. nature, the same result on outliers in Theorem <ref> continues to hold for k_N much smaller than log N, and holds even when k_N is a fixed constant. Also, the assumptions (1), (2), (3) in Theorem <ref> can be considerably weakened: for a square matrix with i.i.d. entries only assuming entries have a finite second moment is sufficient.  These claims are recently proven in <cit.> via the method of characteristic functions, and the proof is adapted from computations in <cit.> and  <cit.>. The results in <cit.> are not accessible by techniques in this paper as this paper relies on a Hermitization procedure, and a Hermitian matrix with only finite second moment entries or constant average degree typically has unbounded operator norm. On the other hand, general band matrices are not accessible by techniques from <cit.> either, as in this case the characteristic function method from <cit.> and  <cit.> breaks down. Also, in our master theorem <ref> we can consider a very general square matrix X without assuming independence of entries, and thus covers many interesting new applications (Theorem <ref>); but the characteristic function method requires strongly that all entries are independent.

For the elliptic case, the assumptions of Theorem <ref> cover (homogeneous) sparse elliptic random matrices, which is the entrywise product of a standard elliptic random matrix as considered in <cit.>, together with a symmetric random matrix B=(b_xy) with B_xy=B_yx∼Ber(k_N/N), for some k_N>>(log N)^5, and that all entries of B are independent modulo symmetry. The conclusion of Theorem <ref> appears to be new even for such (homogeneous) sparse elliptic matrices as the fourth moment of each entry is not bounded, lying outside the assumptions in <cit.>.



A somewhat orthogonal set of assumptions on non-Hermitian random matrices with inhomogeneous profile, is to assume an arbitrary variance profile but with a universal lower and upper bound on the variance of each entry. See <cit.> and <cit.> for local laws and sharp spectral radius bounds in this inhomogeneous case. The assumption that variance of all entries are bounded from below appears to be a crucial one and excludes random band matrices (see <cit.>,<cit.> for a weakening of this hypothesis, but still for dense matrices). 
    Whereas allowing sparsity, our approach instead relies strongly on the regularity hypothesis <ref> (the variance matrix is doubly stochastic) to yield a simple solution to the matrix Dyson equation. To freely change the variance profile of each entry, one needs to find solutions to the matrix Dyson equation with general coefficients, but this seems currently out of reach when the underlying graph G_N is very sparse.





 §.§ Applications: product of independent elliptic matrices
   Random matrix products are a central topic in random matrix theory, and the study of outliers can likewise be generalized to the product setting. Suppose we consider a class of independent non-Hermitian random matrices X_N^i,i=1,⋯,m satisfying Definition <ref>, we can investigate the product matrix (d_N,1)^-1/2X_N^1⋯,(d_N,m)^-1/2X_N^m and its finite dimensional deformation ((d_N,1)^-1/2X_N^1+A_N^1)⋯,((d_N,m)^-1/2X_N^m+A_N^m), which is a multi-matrix version of Theorem <ref>. 



The work <cit.> considered product of independent matrices with i.i.d. entries and showed a similar result as in the one matrix case, that is, the conclusion of Theorem <ref> continues to hold for certain deformations of product i.i.d. matrices. They also mentioned a similar problem: the product of independent elliptic matrices with general ρ_i∈(-1,1). Indeed, <cit.> showed a remarkable universality phenomenon: the empirical spectral distribution of a product of (m≥ 2) independent elliptic random matrices converges to the m-fold circular law, just as in the case of a product of m independent matrices with i.i.d. entries. They also used ideas in free probability to explain why this happens: the product of two free elliptic elements is R-diagonal, and thus has an isotropic Brown measure. However the tools in <cit.> did not seem strong enough to study the outliers of product elliptic matrices due to strong dependence in the entries. 


We show a far-reaching strengthening of the result in <cit.>: exactly the same result continues to hold when we consider independent product of elliptic random matrices with variables ρ_i∈ℂ:|ρ_i|≤ 1, and the conclusion continues to hold as we consider non-Hermitian band matrices as in Theorem <ref> and <ref>. Previously such results were only proven for random matrices with i.i.d. entries.


The main result for product elliptic matrices is the following theorem, which can be regarded as a corollary of our master theorem, Theorem <ref> and <ref>.



 Let m∈ℕ_+ be a fixed integer, and let (d_N,1)^-1/2X_N^1,⋯,(d_N,m)^-1/2X_N^m be independent N× N random matrices satisfying one of the following

    
  * (Independent case) Each (d_N,i)^-1/2X_N^i satisfies the assumption of Theorem <ref>,
 
  * (Elliptic case) Each (d_N,i)^-1/2X_N^i satisfies the assumption of Theorem <ref> with some |ρ_i|≤ 1, and the entries of X_N^i should simultaneously satisfy one of the three constraints on entries in Theorem <ref> or Theorem <ref> (i.e., for i=1,⋯,m they are simultaneously either Gaussian, or bounded, or having bounded p-th moment).Then 
 
  * For any ϵ>0, almost surely as N tends to infinity there is no eigenvalue of 
    D_N^m:=(d_N,1⋯ d_N,m)^-1/2X_N^1
    ⋯ X_N^m
 lying outside of (1+ϵ)𝔻. 
  * Moreover we have the following finite rank perturbation result. Let A_N^1,⋯,A_N^m be deterministic N× N matrices with O(1) rank and operator norm. Then consider 
    D_N^m,1:=∏_k=1^m ((d_N,k)^-1/2X_N^k+A_N^k),   A_N:=∏_k=1^m A_N^k.
 Fix an ϵ>0 and assume that there is no eigenvalue of A_N lying in {z∈ℂ:1+ϵ≤|z|≤ 1+3ϵ} and that there are j=O(1) eigenvalues λ_1(A_N),⋯,λ_j(A_N) outside (1+3ϵ)𝔻. Then almost surely as N tends to infinity, there are exactly j eigenvalues λ_1(D_N^m,1),⋯,λ_j(D_N^m,1) lying outside (1+2ϵ)𝔻 and after rearrangement, we have
    
    λ_i(D_N^m,1)=λ_i(A_N)+o(1),   i=1,⋯,j.

    


The special case m=2 and ρ_1=ρ_2=1 implies the eigenvalues of the product of two i.i.d. Wigner matrices are contained in (1+ϵ)𝔻 with high probability for any ϵ>0.


    For any non commutative polynomial P with m+n non commutative variables, one may consider the outliers of P evaluated at m independent random matrices and n deterministic matrices. One also assumes the n deterministic matrices have a strong limit in free probability. For random matrices with i.i.d. entries this was investigated in <cit.>. For non-Hermitian band matrices, it would be possible to generalize Theorem <ref> to outliers of any polynomial P, but this requires considerably more effort and is left for further study.





 §.§ Proof outline

Now we briefly discuss the context of the problem investigated, and illustrate the major difficulties and key proof strategies.

For random band matrices with independent entries with bandwidth d_N (a special case of Definition <ref>), a major open problem (see <cit.>,  <cit.>, <cit.>) is to prove the convergence of the empirical spectral density to a deterministic limit. The limit should be the circular law as long as d_N grows with N, which has been proved for sparse i.i.d matrices in <cit.>. The major technical difficulty to proving the circular law (see for example <cit.>) is to prove an estimate on the least singular value of zI_N-(d_N)^-1/2X_N for a.e. z∈ℂ, so that Girko's Hermitization technique <cit.> <cit.> can be rigorously justified. This step is highly model specific and the proof techniques for i.i.d. sparse matrix <cit.> do not generalize to arbitrary band matrices which has much less homogeneity. Recently, <cit.> and <cit.> introduced new methods to prove the circular law for periodic block band matrices and periodic band matrices for bandwidth d_N>>N^33/34. Yet the case of smaller d_N still seems to be open.

The spectral radius for inhomogeneous random matrices is much easier to study thanks to the method of moments. Via computing the trace of a sufficiently large ℓ∼log N power of (d_N)^-1/2X_N, the spectral radius of (d_N)^-1/2X_N can be bounded with very high probability. In <cit.>, Theorem 2.11 and Remark 2.13, a method is introduced to bound the spectral radius for inhomogeneous non-Hermitian random matrix that covers (d_N)^-1/2X_N when d_N≥ N^ϵ for any ϵ>0. In Theorem <ref> we illustrate the same spectral radius bound can be achieved whenever d_N=(log N)^5. We note that when g_xy are standard (real) Gaussian variables, <cit.>, Theorem 3.1 proved that (d_N)^-1/2X_N_op=2(1+o(1)) with high possibility whenever d_N>>log N. This result is optimal in the sparsity of d_N and implies ρ((d_N)^-1/2X_N)=2(1+o(1)), but the operator norm bound loses a factor 2 compared with the spectral radius bound 1+o(1).


The problem of determining the spectral outliers of (d_N)^-1/2X_N+C_N for some finite rank perturbation C_N, does not appear to be studied before in such an inhomogeneous setting, and is the main topic of the current work. On one hand, finding spectral outliers are more tractable than proving the empirical density convergence: We do not need to estimate the least singular value, which is a notoriously difficult problem. On the other hand, it appears that all existing methods on this topic require strong homogeneity on the variance profile of X_N. The first work for non-Hermitian matrices <cit.> made use of a moment comparison to the Ginibre ensemble, and later works either followed a similar line, or resorted to proving an isotropic law (see <cit.>) whose proof does not carry over to general inhomogeneous matrices X_N with d_N=o(N). The method of moments, or some modifications of it, might be a promising route to study outliers of (d_N)^-1/2X_N+C_N. However, as the eigenvalues are complex, its distribution is not completely determined by the moments. The method of characteristic functions can alternatively be used to detect spectral outliers, see <cit.>. However, for this method to work we need to prove joint convergence of traces of (d_N)^-1/2X_N to a family of independent Gaussian variables, which is simply not the case in the current setting of inhomogeneous random matrix on regular graphs. Indeed, the characteristic function (I_N-z(d_N)^-1/2X_N) does not form a tight family of holomorphic functions for |z|≤ 1-ϵ, so that the method breaks down completely. Relevant moment computations for band matrices, also called genus expansion when the entries are Gaussian distributions, can be found in <cit.> for the i.i.d. case and <cit.> for the Hermitian case. These genus expansion formulas (asymptotic freeness up to first order) are only effective for d_N>>N^1/3 and d_N>>N^1/2 respectively, falling short of the smaller d_N case. Therefore, we see that genus expansion is ineffective for d_N=o(N^1/3) and the characteristic function technique is ineffective for any d_N=o(N), and completely new techniques are needed here.

We will first take a standard Hermitization procedure and consider for any z∈ℂ, 
    𝒴_z:=[        0   X-zI_N; X̅-z̅I_N       0, ].

 We need to show that σ_min(𝒴_z)>0 when z is away from the unit disk (or ellipse ℰ_ρ), and prove an isotropic law for 𝒴_z for z in the same region. The inhomogeneous sparse variance pattern of X makes standard techniques ineffective, and we will make use of recent advances in free probability and universality principles for highly inhomogeneous self adjoint random matrices, which are illustrated in <cit.> for the Gaussian case and <cit.> for the general case. The idea is to consider a free probability model 𝒴_z,free defined from 𝒴_z, and the assumptions on X in Theorem <ref> (the variance profile being doubly stochastic and 𝔼[X^2] is a multiple of identity) make the spectral properties of 𝒴_z,free remarkably transparent. Then universality principles in the cited papers enable us to transfer the estimates back to our random matrix X. Similar ideas have appeared in <cit.> which also partially inspires our work.
 

The main result of this paper can be seen as a non-Hermitian generalization of recent studies of spectral outliers for random band matrices. For periodically banded Hermitian matrices, the finite rank perturbation properties (the BBP transform introduced in <cit.>) are recently studied in <cit.>. Later <cit.> introduced a very general framework for solving such problems in the self-adjoint setting, and established the BBP type phenomena for many other inhomogeneous random matrices with divers patterns of variance profiles. Theorem <ref> can be thought of as a non-Hermitian analogue of <cit.>, Theorem 3.1. In contrast to the Hermitian case, we do not discuss the eigenvector overlaps associated to the outlying eigenvalues. While this might theoretically be possible to investigate, a closed form expression is too difficult to work out.





§ PREPARATIONS: FREE PROBABILITY AND UNIVERSALITY THEOREMS





 §.§ Free probability preliminaries

We first recall some standard facts in free probability that can be found in <cit.>.
Consider a C^* probability space (𝒜,^*,τ,·), where (𝒜,^*,·) is a C^*-algebra and τ is a faithful trace. As τ is a faithful trace, the norm · is uniquely determined by (<cit.>, Proposition 3.17)

    a=lim_k→∞(τ[(a^*a)^k])^1/2k,  ∀ a∈𝒜.



A family of self-adjoint elements s_1,⋯,s_m∈𝒜 are called a free semicircular family  if for all p≥ 1 and k_1,⋯,k_p∈ [m] we have

    τ(s_k_1⋯ s_k_p) =∑_π∈NC_2([p])∏_{i,j}∈πδ_k_i,k_j,

where NC_2([p]) is the set of non-crossing partitions of [p]. 





 §.§ Universality principles





For the Gaussian case, let X be a Gaussian random matrix admitting a decomposition (<ref>). Although in the statement of Theorem <ref> the matrix X is non-Hermitian, we need to consider Hermitian matrices when stating the spectral universality theorems. This will be achieved via a standard Hermitization procedure. Thus in this section we assume the matrix (<ref>) is Hermitian and A_0,⋯,A_n∈ M_N(ℂ)_sa, where M_N(ℂ)_sa denotes the set of N by N square matrices which are complex valued and self-adjoint.

To the Gaussian model (<ref>)
we introduce the following free element X_free associated to X:


    X_free:=A_0⊗1+∑_i=1^n A_i⊗ s_i,
where A_0,⋯,A_n∈ M_N(ℂ)_sa and s_1,⋯,s_d a free semicircular family. 




For a general matrix model X as defined in (<ref>), we associate X with a Gaussian model G of N× N matrix such that {ReG_ij,ImG_ij:i,j∈[N]} are jointly Gaussian, that 𝔼[G]=𝔼[X] and that Cov(G)=Cov(X). Then we associate to X a free model X_free via setting 
    X_free:=G_free,
 where G_free is the free model associated to the Gaussian model G as defined in (<ref>). We remark that for a Gaussian matrix X, there could be multiple ways to write X as a sum in the form of (<ref>), so there could be multiple ways to define X_free. It turns out that any such X_free will be sufficient for our use.


For the Gaussian model, we have the following concentration and universality results:


(Resolvent convergence, Gaussian case, <cit.>, Theorem 2.8)
Let A_0,⋯,A_n∈ M_N(ℂ)_sa. The matrix valued Stieltjes transform for Z∈ M_d(ℂ) is defined as

    G(Z):=𝔼[(Z-X)^-1],   G_free(Z):=(id⊗τ))[(Z⊗1-X_free)^-1].

    Then we have the estimate
    
    G(Z)-G_free(Z)≤ṽ(X)^4(ImZ)^-5
 which holds for any Z∈ M_d(ℂ) such that ImZ:=1/2i(Z-Z^*)>0.



For the Gaussian case we only need the following concentration estimate, which is very similar to <cit.>, Lemma 5.5. The proof is given in Appendix <ref>.
(Resolvent concentration, Gaussian case) Fix any two unit vectors u,v∈ℝ^N. Fix z∈ℂ with Imz>0. For any x>0 we have
    
    ℙ[| ⟨ u,
    (z1-X)^-1v⟩-𝔼⟨ u,(z1-X)^-1v⟩|≥σ_*(X)/(Imz)^2
    x]≤ 2e^-x^2/2.



For any self-adjoint operator X we let sp(X)⊂ℝ denote the spectrum of the operator X. We have

(Spectrum concentration, Gaussian case, <cit.> Theorem 2.1)
    For the model (<ref>) with A_0,⋯,A_n∈ M_N(ℂ)_sa, then for all t>0,
    
    ℙ[ sp(X)⊆sp(X_free)+C{ṽ(X)(log N)^3/4+σ_*(X)t}[-1,1
    ]]≥ 1-e^-t^2,
 for a universal constant C>0.


For the general case, we will need similar convergence and concentration results. We have the following universality principle:

(General entry resolvent university, <cit.> Theorem 2.11) 

For every z∈ℂ such that Im(z)>0, we have

    𝔼[(z1-X)^-1]-𝔼[(z1-G)^-1]≲σ(X)+R̅(X)^1/10σ(X)^9/10/(Im(z)^2.



We will also need the following concentration result on resolvent entries, that will be proved in Section 4. The proof is outlined in Appendix <ref> and is very similar to the argument in <cit.>, Proposition 5.6.




(General entry resolvent concentration) 
For every z∈ℂ such that Im(z)>0, and any two unit vectors u,v∈ℝ^N with u_L^2=v_L^2=1, we have

    ℙ[|⟨ u,(z1-X)^-1v⟩-𝔼⟨ u,(z1-X)^-1v⟩|≥σ(X)/(Im(z)^2)√(x)+{R(X)/Im(z)^2+R(X)^2/Im(z)^3}x
            +{R(X)^1/2(𝔼X-𝔼X)^1/2/Im(z)^2+R(X)(𝔼X-𝔼X^2)^1/2/Im(z)^3}√(x)
    ]≤ 2e^-Cx

for a universal constant C>0 and for any given x≥ 0.



(General entry spectrum concentration, <cit.> Theorem 2.6) For the model (<ref>) with Z_0,⋯,Z_n∈ M_N(ℂ)_sa, such that Z_i≤ R a.s. for each i=1,⋯,n. Then we have for all t≥ 0, 
       
    ℙ[ sp(X)⊆sp(X_free)+C{σ_*(X)t^1/2+R^1/3σ(X)^2/3t^2/3+Rt}[-1,1]
    ]≥ 1-Ne^-t^2,

    where C>0 is a universal constant.






§ PROOF OF MAIN RESULTS



We first prove the very simple lemma Lemma <ref>. We claim no novelty for its proof as can be found in (4.6)-(4.8) of <cit.>.

  First assume that ρ is real, then taking the real and imaginary parts of z=x+ρx̅ implies Rez=Rex+ρRex and Imz=Imx-ρImx. Then since |x|≤ 1, we have 
    Re(z)^2/(1+ρ)^2+Im(z)^2/(1-ρ)^2≤ 1.
 All these relations can be reversed, hence proving equivalence.

For general ρ∈ℂ, we have that z=x+ρx̅ if and only if 
    e^-i(ρ)/2z=e^-i(ρ)/2x+|ρ|e^-iarg(ρ)/2x
 for any x∈ℂ:|x|≤ 1. This implies the equivalence.    



Before entering the proof we introduce elliptic random variables in free probability, which generalizes the semicircular and circular variables introduced by Voiculescu <cit.>.

We shall also use some notations on operator-valued free probability theory introduced in <cit.> and we also refer to <cit.>, Section 3.2.1 for some basic notations. 



    In a C^*-non-commutative probability space (𝒜,τ), an element c is called a circular element if c=1/√(2)s_1+i/√(2)s_2 for two free semicircular variables s_1,s_2. 
    
    
    
    In general, for any ρ∈[-1,1] we call c_ρ:=√(1+ρ/2)s_1+i√(1-ρ/2)s_2 an elliptic element of parameter ρ where s_1,s_2 are two free semicircular variables.





 §.§ No outliers in Theorem <ref>
  
We start with proving the master theorem in Gaussian case, Theorem <ref>. 


First we need to analyze the free operator X_free associated to X. Fix any z∈ℂ and consider the Hermitization of X at z: define

    𝒴_z:=[        0   X-zI_N; X̅-z̅I_N       0, ]
 and in particular when z=0 we write 

    𝒴=[  0  X; X̅  0 ].

The associated free model can be written as 

    𝒴_z,free=[      0  -zI_N; -z̅I_N      0 ]+[       0  X_free; X̅_free       0 ].
 



Also, for any v=E+iη∈ℂ with η>0, we define

    𝒴_z(v)=[ -v I_N  -zI_N; -z̅I_N -v I_N ]+[   0  X_; X̅_   0 ],  and


    𝒴_z,free(v)=[ -v I_N  -zI_N; -z̅I_N -v I_N ]+[       0  X_free; X̅_free       0 ].



To compute the Stieltjes transform we need the following computation. For a self-adjoint random matrix A=A_0+∑_i=1^nA_ig_i where A_i∈ M_N(ℂ)_sa are self adjoint and g_i are independent real Gaussians of mean 0, variance 1, then for any M∈ M_N(ℂ)_sa, we have


    ∑_i=1^n A_iMA_i=𝔼[(∑_i=1^n A_ig_i)M(∑_i=1^n A_ig_i)]=𝔼[(A-𝔼A)M(A-𝔼A)].



We define the Stieltjes transform of the free model 𝒴_z,free as follows: for any v∈ℂ_+:={z=E+iη∈ℂ:η>0}, we define

    G_z(v):=(id⊗τ)(𝒴_z,free(v)^-1).



By <cit.>, equation 1.5 and the computation (<ref>), the Stieltjes transform G_z(v) solves the following self consistency equation

    𝔼[𝒴G_z(v)𝒴]+G_z(v)^-1+[ v I_N   z I_N; z̅I_N   v I_N ]=0.
 By <cit.>, Theorem 2.1, for any v∈ℂ_+ there is a unique solution G_z(v) to this equation such that G_z(v) has a positive imaginary part. Recall that for a square matrix M, its imaginary part is defined as ImM=1/2i(M-M^*).

By symmetry we may assume (and will be proved later) that the unique solution to (<ref>) with positive imaginary part is given by 

    G_z(v)=[ a(z,v) I_N  b(z,v)I_N; b̅(z,v)I_N  c(z,v)I_N ],
 where a(z,v), b(z,v) and c(z,v) are scalar functions depending on z and v (as we ll as ρ and D_N, which are suppressed from notation). 

Taking this block-diagonal form into (<ref>), the equation transforms to


    [        c (ρ+D_N)b; (ρ+D_N)b        a ]
    +
        1/ac-|b|^2[   c  -b; -b̅   a ]+[  v  z; z̅  v ]=0.


We now show that (<ref>) has a unique solution such that [  a  b; b̅  c ] has a positive imaginary part, and that the solution satisfies some desired properties. We do not try to solve (<ref>) directly, but we show another (well-studied) object in free probability has the same Stieltjes transform.

For this we denote by ρ_0:=ρ+D_N and set θ_0:=ρ_0. Note that by assumption we have |ρ+D_N|≤ 1. We consider an elliptic element c_|ρ_0| multiplied by e^iθ_0/2 and embed it into a two by two matrix (where s_1 and s_2 two free semicircular variables), for any z∈ℂ,

    𝒞_z=[                                                  0  e^iθ_0/2(√(1+|ρ_0|/2) s_1+i√(1-|ρ_0|/2) s_2)-z1_𝒜; e^iθ_0/2(√(1+|ρ_0|/2) s_1-i√(1-|ρ_0|/2) s_2)-z̅1_𝒜                                                  0 ].


We claim that the Stieltjes transform of 𝒞_z, defined as 

    𝒢_z(v):=(id⊗τ
    )((𝒞_z-vI_2)^-1):=[  a(z,v)  b(z,v);  b(z,v) c(z,v), ]
for v∈ℂ_+, satisfies that (a,b,c) solves the same equation as (<ref>) if we substitute a=a, b=b and c=c in equation (<ref>). This can be verified via directly applying <cit.>, equation 1.5, so that (a,b,c) should solve the same equation as (<ref>). 


The matrix 𝒢_z(v) has positive imaginary part as it is the stieltjes transform of 𝒞_z, so that the existence (and uniqueness) to (<ref>) with positive real part is now well-established.

By Riesz representation theorem, we can find unique probability measures μ_1,μ_2 on the real line such that for any φ∈ C_0(ℝ),

    ∫_ℝφ dμ_1=(tr⊗τ)φ(𝒞_z),


    ∫_ℝφ dμ_2=(tr⊗τ)φ(𝒴_z,free).

Let g_1(v):=(tr⊗τ)((𝒞_z-vI_2)^-1) and  g_2(v):=(tr⊗τ)((𝒴_z,free(v))^-1). Then g_1=g_2 by our previous discussion, and by Stieltjes inversion formula 

    μ_i=lim_y→ o^+(1/πIm(g_i(x+iy))dx),
 we identify that μ_1=μ_2. Since τ is faithful, we may choose φ supported on a neighborhood of the complement of the spectrum of 𝒞_z (or 𝒴_z,free), and deduce that 
    sp(𝒞_z)=sp(𝒴_z,free).


Now we determine sp(𝒴_z,free) from sp(𝒞_z) as the latter is the rotation of an elliptic element. Indeed, 𝒞_0 is the self-adjoint dilation of an elliptic element c_|ρ_0| rotated by θ/2, and the spectral support of an elliptic element c_|ρ_0| has been computed, see for example <cit.>. In particular <cit.> implies that for any z∈ℂ∖ℰ_|ρ_0|,ϵ and any ϵ>0, we have that σ_min(c_|ρ_0|-z1_𝒜)>0. 


By definition, ℰ_ρ_0 for complex ρ_0 is the rotation of ℰ_|ρ_0| by e^iθ_0/2, so that for any z∈ℂ∖ℰ_ρ_0,ϵ and any ϵ>0, we have that σ_min(e^iθ_0/2c_|ρ_0|-z1_𝒜)>0. Using the fact that σ_min(e^iθ_0/2c_|ρ_0| -z1_𝒜) is Lipschitz continuous in z and taking a finite covering over {z∈ℂ:|z|≤ 10}∖ℰ_ρ_0,ϵ), we deduce that we can find constant C_ρ,ϵ>0 depending only on ρ,ϵ such that 
inf_z∈ℂ∖ℰ_ρ,ϵσ_min(e^iθ_0/2c_|ρ_0|-z1_𝐀)≥ C_ρ,ϵ>0. In the last step we have used two reductions: first, using the näive estimate c_|ρ_0|≤ 6, then for any |z|>10 the nonzero lower bound on σ_min is trivial; and second, as D_N→ 0 we may assume N is large enough such that dist(ℰ_ρ,ϵ,ℰ_ρ_0,ϵ)≤ϵ/10. Rearranging the value of ϵ justifies the claim.   

In the last paragraph, we have justified that (whenever N is large) for any z∈ℂ∖ℰ_ρ,ϵ we have [-C_ρ,ϵ,C_ρ,ϵ]∉sp(𝒞_z). By previous identification (<ref>) we then have 
    [-C_ρ,ϵ,C_ρ,ϵ]∉sp(𝒴_z,free).
 Now we apply universality principles to show the same holds for sp(𝒴_z) with high probability, justifying that X has no outliers:


 Now the proof of no-outliers is a simple application of Theorem <ref> together with estimate (<ref>). We only need to consider the region |z|≤ 10, as the main results of <cit.> already imply that X_op≤ 6 with very high probability. For each fixed z∈ℂ∖ℰ_ρ,ϵ we apply Theorem (<ref>) to deduce that, with probability at least 1-C_1N^-1.5, [-C_ρ,ϵ/2,C_ρ,ϵ/2]∉sp(𝒴_z), so that σ_min(X-zI_N)≥ C_ρ,ϵ/2>0 with probability at least 1-C_1N^-1.5. Since σ_min(X-zI_N) is Lipschitz continuous in z, we can take a finite covering of {z∈ℂ:|z|≤ 10} and take a union bound to conclude that with probability at least 1-C_1N^-1.5, we have 
    inf_z∈ℂ∖ℰ_ρ,ϵσ_min(X-zI_N)≥ C_ρ,ϵ/2>0.
 
    The constant C_1 may change from line to line but it depends only on ρ and ϵ.





 §.§ Isotropic laws


In this section we determine an isotropic law, i.e. an almost sure limit of ⟨ u,(X-zI_N)^-1v⟩ for unit vectors u,v and z∈ℂ∖ℰ_ρ,ϵ in the setting of Theorem <ref>.

In the following we only need to evaluate the Stieltjes transform 𝒴_z,free(v) at v=iη for any η>0, and we will pass to the limit η↘ 0 in the end.


Recall that provided X-zI_N is invertible, we shall have 
    𝒴_z(0)^-1=[             0 (X̅-z̅I_N)^-1;   (X-zI_N)^-1            0. ].
 


For any z∈ℂ∖ℰ_ρ,ϵ, on the event where (<ref>) happens (which has possibility at least 1-C_1N^-1.5 under the assumption of Theorem <ref>), we have that 

    𝒴_z(0)^-1_op≤ 2/C_ρ,ϵ<∞

is uniformly bounded for fixed ϵ.
Then from the resolvent identity 
    (λ I-A)^-1-(λ I-B)^-1=(λ I-A)^-1(A-B)(λ I-B)^-1

we can deduce that for η>0 sufficiently small one must have

    (𝒴_z(iη)^-1_op≤(𝒴_z(0)^-1_op+(𝒴_z(iη)^-1_opη(𝒴_z(0)^-1_op,
 so that for η∈[0,C_ρ,ϵ/4], we have that on the event where (<ref>) happens,

    (𝒴_z(iη)^-1_op≤ 2(𝒴_z(0)^-1_op≤ 4/C_ρ,ϵ.
 

Similarly, for any z∈ℂ∖ℰ_ρ,ϵ
we already have proved in (<ref>) that 

    𝒴_z,free(0)^-1_op≤ 1/C_ρ,ϵ<∞.

In the following we will compute lim_η↘ 0(𝒴_z,free(iη))^-1 and use the universality principle in Section <ref> to compute 𝒴_z(0)^-1.

We set v=iη in the self-consistency equation (<ref>). By symmetry, the solution should satisfy a=c (this can be checked by noting that, swapping the role of a and c, the pair (c,b,a,d) is again a solution provided that (a,b,c,d) is a solution, and the new solution still has positive definite imaginary part; then we resort to uniqueness of solutions in <cit.>). Then the individual coordinate equations are given by  


    a(1+1/a^2-|b|^2)+iη=0,
 



    b̅(ρ+D_N)-b/a^2-|b|^2=-z.






Next we show that a=c is purely imaginary and has a positive imaginary part. We will prove this by showing that there exists a solution to (<ref>) with purely imaginary a that also satisfies the positive definite imaginary part assumption, so that by uniqueness <cit.> the solution must be of this form. We take 
    a(z,η)=iV(z,η),   V(z,η)∈ℝ_+.
 Then the equation simplifies to 

    V(1-1/|V|^2+|b|^2)+η=0,  b̅(ρ+D_N)+b/|V|^2+|b|^2+z=0.


    For any η>0, any z∈ℂ and |ρ+D_N|≤ 1 there exists a solution V>0 to (<ref>).

 We may without loss of generality assume ρ_0:=ρ+D_N is real (and recall θ_0=ρ_0), this is because for a solution (V,b) to (<ref>) with parameter (ρ_0,z) replaced by  |ρ_0| and e^-iθ/2z, via a simple calculation we see that (V,e^iθ_0/2b) is a solution to (<ref>) with parameter ρ_0 and z. 

Then in the case where ρ_0 is real, we can follow computations as in <cit.>, Lemma C.2 to see that V solves the following equation (we have changed notation a↦ iV and η→ iη in <cit.>, C.2):
    
    1-1/(V+η)V=-Re(z)^2/(η+(1+ρ_0)V)^2-Im(z)^2/(η+(1-ρ_0)V)^2
 Setting V→ 0+ and setting V→ +∞ on both sides of the equation shows that a solution must exist.


Having checked a is purely imaginary, we now derive η↘ 0 asymptotics of a.
In the following we consider z∈ℂ∖ℰ_ρ,ϵ. Suppose that limsup_η↘ 0|a(z,η)|>0, then on a sequence η_n↘ 0 we have |V|^2+|b|^2→ 1. Together with the fact that D_N→ 0 as N→ 0, we see that whenever N is large enough, this forces 
    b+b̅ρ=-z+o(1)
 on this sequence η_n. The asymptotic |V|^2+|b|^2→ 1 forces |b|≤ 1+o(1) Combining this with (<ref>) and recalling the definition of ℰ_ρ,ϵ, this leads to a contradiction to the fact that z∈ℂ∖ℰ_ρ,ϵ 


Thus for any fixed ϵ>0 and any z∈ℂ∖ℰ_ρ,ϵ, whenever N is sufficiently large,  we must have lim_η↘ 0|V(z,η)|=0. Via a standard computation we deduce that 
    lim_η↘ 0b̅(z,η)=-z+√(z^2-4ρ_0)/2ρ   ρ_0≠ 0
    -1/z   ρ_0=0
 where we take the square root √(z^2-4ρ_0) with branch cut on the segment connecting -2√(ρ_0) and 2√(ρ_0) in the complex plane and such that lim_z→∞√(z^2-4ρ_0)-z=0. 

We have proved the following: for any fixed ϵ>0, whenever N is large enough, we have

    lim_η↘ 0
         a(z,η)=0,  lim_η↘ 0b̅(z,η)=-z+√(z^2-4ρ_0)/2ρ   ρ_0≠ 0,
    
        -1/z   ρ_0=0.


Now we are ready to proving the following isotropic law, which is the cornerstone to Theorem <ref> and generalizes <cit.>, Theorem 5.1.

 (Isotropic law)
    In the setting of Theorem <ref>, set δ_N=(loglog N)^-1. Then for any unit vectors u,v∈ℝ^N, with probability at least 1-C_1N^-1.4, we have

    sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    (X-zI_N)^-1v⟩ --z+√(z^2-4ρ)/2ρ⟨ u,v⟩|≤ C_2δ_N,  ρ≠ 0,
    
       sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    (X-zI_N)^-1v⟩ +1/z⟨ u,v⟩|≤ C_2δ_N,  ρ=0,
 where C_1,C_2>0 are some universal constants depending only on ρ and ϵ. Also, the same estimate holds if we replace δ_N by (δ_N)^p for any p>0.




We invoke Theorem <ref> to deduce that for any η>0 and z∈ℂ,

    𝔼[(𝒴_z(iη)^-1-(id⊗τ)𝒴_z,free(iη)^-1]≤(log N)^-1/η^5.
 
We then invoke Theorem <ref> to deduce that for any η>0 and z∈ℂ, for any two unit vectors u,v∈ℝ^2N, we have (taking x=c√(log N) in Theorem <ref> for a large c>0)

    ℙ( | ⟨ u,(
    𝒴_z(iη)^-1v⟩ -𝔼⟨ u,𝒴_z(iη)^-1v⟩|≥(log N)^-1/4/η^2))≤ 2N^-10.


In the following we take 
    η=(loglog N)^-1.
  By our choice of η and resolvent identity (<ref>), we can easily check that the function z↦⟨ u, 𝒴_z(iη)^-1v⟩ is Lipschitz continuous in z∈ℂ∖ℰ_ρ,ϵ with a Lipschitz constant (loglog N)^2. Therefore taking a covering of z∈ℂ∖ℰ_ρ,ϵ of mesh size 10^-2δ_N(loglog N)^-2 and then apply a union bound to estimate (<ref>) for z ranging over these mesh points and |z|≤ T_N:=100loglog N, and then use the (loglog N)^2- Lipschitz continuity of the function z→⟨ u,𝒴_z(iη)^-1v⟩ in z, we deduce that 

    ℙ( sup_z∈(ℂ∖ℰ_ρ,ϵ)∩{z∈ℂ:|z|≤ T_N}| ⟨ u,
    𝒴_z(iη)^-1v⟩ -𝔼⟨ u,𝒴_z(iη)^-1v⟩|≥δ_N)= O( N^-9).

 Since 𝒴_0(iη)≤ 6 for η>0 small with probability 1-O(N^-1.4) (this can be proven via applying <cit.>, Corollary 2.2),  we can ensure 𝒴_z(iη)
^-1≤1/4δ_N for |z|>T_N and therefore we remove the constraint |z|≤ T_N and get,

    ℙ( sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    𝒴_z(iη)^-1v⟩ -𝔼⟨ u,𝒴_z(iη)^-1v⟩|≥δ_N)= O( N^-1.4).


Applying a similar continuity argument to (<ref>) we get 


    sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,(id⊗τ)𝒴_z,free(iη)^-1
    v⟩ -𝔼⟨ u,𝒴_z(iη)^-1v⟩|≤δ_N.



Combined with (<ref>), we get 

    ℙ( sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    𝒴_z(iη)^-1v⟩ -⟨ u,(id⊗τ)𝒴_z,free(iη)^-1
    v⟩|≥ 2δ_N)= O( N^-1.4).




Now we set up a different estimate.
Consider the following event 
    Ω_N(z):={𝒴_z(iη)^-1_op≤ 4/C_ρ,ϵ for all η∈[0.C_ρ,ϵ/4]}.
 Since z∈ℂ∖ℰ_ρ,ϵ, using the estimate (<ref>) we have ℙ(Ω_N(z))≥ 1-O(N^-1.5). Since z↦𝒴_z(iη)^-1_op is (loglog N)^2-Lipschitz continuous, we take a covering of (ℂ∖ℰ_ρ,ϵ)∩{|z|≤ T} of mesh size 10^-2/C_ρ,ϵ(loglog N)^-2  to upgrade the bound to be uniform over z∈ℂ∖ℰ_ρ,ϵ (the bound 𝒴_z(iη)^-1_op≤ 4/C_ρ,ϵ for |z|>T is trivial when T large ), and conclude with the following: consider the event

    Ω_N:={sup_z∈ℂ∖ℰ_ρ,ϵ,η∈[0.C_ρ,ϵ/4]𝒴_z(iη)^-1_op≤ 4/C_ρ,ϵ},

then 
    ℙ(Ω_N)≥ 1-O(N^-1.4).
 Moreover, by the resolvent identity (<ref>), on the event Ω_N the mapping η↦𝒴_z(iη)^-1_op is Lipschitz continuous in η>0 with a Lipschitz constant (4/C_ρ,ϵ)^2 uniformly in z∈ℂ∖ℰ_ρ,ϵ. That is, we have that for N sufficiently large,

    𝒴_z(iη)^-1-𝒴_z(0)^-1_op≤ (16/C_ρ,ϵ^2)δ_N for all z∈ℂ∖ℰ_ρ,ϵ   on Ω_N.


Finally, the asymptotic in (<ref>) for entries of the free operator, and the fact that the function z↦(id⊗τ)𝒴_z,free(iη)^-1
is also (4/C_ρ,ϵ)^2-Lipschitz continuous for z∈ℂ∖ℰ_ρ,ϵ (which follow from the fact that σ_min(𝒴_z,free) is bounded away from zero throughout z∈ℂ∖ℰ_ρ,ϵ) imply via a continuity argument that

    sup_z∈ℂ∖ℰ_ρ,ϵ(id⊗τ)𝒴_z,free(iη)^-1
    -[           0  b(z,0+)I_N; b̅(z,0+)I_N           0 ]_op≤(16/C_ρ,ϵ^2)δ_N,
 where we again take the choice η=(loglog N)^-1, and the expression b(z,0+) is given in (<ref>).

We can now conclude the proof via combining (<ref>), (<ref>) and (<ref>). Combining these estimates we have: for some universal constants C_1,C_2>0 depending on ρ,ϵ,

    ℙ( sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    𝒴_z(0)^-1v⟩ -⟨ u,[           0  b(z,0+)I_N; b̅(z,0+)I_N           0 ]
    v⟩|≥ C_2δ_N)≤ C_1N^-1.4.


Recall that 𝒴_z(0)^-1 has the expression (<ref>). Thus, for any unit vector u,v∈ℝ^N, setting (u,o) and (0,v) to be the two unit vectors in ℝ^2N in the previous estimate, we derive the (final) isotropic estimate for the original non-Hermitian matrix X: 

    ℙ( sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    (X-zI)^-1v⟩ --z+√(z^2-4ρ)/2ρ⟨ u,
    v⟩|≥ C_2δ_N)≤ C_1 N^-1.4, for ρ≠ 0
    
       ℙ( sup_z∈ℂ∖ℰ_ρ,ϵ| ⟨ u,
    (X-zI)^-1v⟩ +1/z⟨ u,
    v⟩|≥ C_2δ_N) ≤ C_1 N^-1.4, for ρ= 0.

In the last step we have used  (<ref>) and the assumption that ρ=ρ_0-D_N and that  lim_N→∞(log N)D_N=0.
Going through the above proof, we see that exactly the same argument works if we replace δ_N by (δ_N)^p and take η=(loglog N)^-p for any p>0.







 §.§ Determination of outliers

Finally we complete the determination of outlying eigenvalues, which completes the proof of Theorem <ref>. 


For this purpose we will do the singular value decomposition for the deterministic matrix C_N: 

    C_N=A_AB_N

where A_n is some N× k matrix and B_N is some k× N matrix, both of which having a bounded operator norm.

Following <cit.>, Lemma 2.1, we have the following eigenvalue criterion: a complex z∈ℂ is an eigenvalue of X+C_N but is not an eigenvalue of X if and only if 

    (1+B_N(X-zI_N)^-1A_N)=0.

The proof is a standard linear algebra exercise using the matrix identity 

    (1+AB)=(1+BA)

for any n× k matrix A and k× n matrix B.
Now we conclude the proof of Theorem <ref>.





Define 
    f(z)=(1+B_N(X-zI_N)^-1A_N)

and 

    g(z)=(1+B_N-z+√(z^2-4ρ)/2ρA_N),  ρ≠ 0,
       g(z)=(1+B_N(-1/z)A_N),  ρ=0.


By a standard exercise in complex analysis, the function z↦-z+√(z^2-4ρ)/2ρ bijectively maps ℂ∖ℰ_ρ onto 𝔻=ℰ_0 when ρ≠ 0. The same holds for z→-1/z when ρ=0.

Then using the matrix identity (<ref>) we deduce that the roots of g(z) on ℂ∖ℰ_ρ are precisely given by λ_i(C_N)+ρ/λ_i(C_N),1≤ i≤ j, where 
 λ_1(C_N),⋯,λ_j(C_N) are the eigenvalues of C_N such that |λ_i(C_N)|≥ 1 for each i, so there are only finitely many of them. 
 
 We first assume all the roots of g in ℂ∖ℰ_ρ,ϵ are simple roots.
 From the isotropic law proved in Theorem <ref>, we claim that on an event with probability at least 1-C_1N^-1.4,
for each nonzero eigenvalue λ_i(C_N) such that |λ_i(C_N)|>1,λ_i(C_N)+ρ/λ_i(C_N)∈ℂ∖ℰ_ρ,ϵ, 

    |f(z)-g(z)|<|g(z)|, for all z∈∂ B(λ_i(C_N)+ρ/λ_i(C_N),(δ_N)^3/4),
 
where for any x∈ℂ and δ>0 we let B(x,δ) denotes the disk with center x and radius δ in the complex plane, and ∂ B(x,δ) denotes its boundary. To see why this is true, we assume N is large enough so that g has no other roots in this disk B(λ_i(C_N)+ρ/λ_i(C_N),(δ_N)^3/4) so that (as g has simple roots) |g(z)|∼ (δ_N)^3/4 on ∂ B(λ_i(C_N)+ρ/λ_i(C_N),(δ_N)^3/4)
and implies the bound (<ref>) thanks to the isotropic law
in Theorem <ref> and that (δ_N)^3/4>>δ_N.


Hence by Rouche's theorem in complex analysis, f(z) has a solution that is (δ_N)^3/4- close to λ_i(C_N)+ρ/λ_i(C_N) (i.e. lies in B(λ_i(C_N)+ρ/λ_i(C_N),(δ_N)^3/4)) for each i=1,⋯,j.  Likewise, fix another domain B(x,δ)⊂ℂ∖ℰ_ρ,ϵ on which g has no zeros, we must have |f(z)-g(z)|<g(z) for z∈∂ B(x,δ) when N is large, hence by Rouche's theorem f has no zeros on B(x,δ). This characterizes all roots of f in ℂ∖ℰ_ρ,ϵ and hence determines all outlying eigenvalues of X+C_N in ℂ∖ℰ_ρ,ϵ. This completes the proof of Theorem <ref> as we have (δ_N)^3/4=o((loglog N)^-1/2).


When g has repeated roots, note that g has degree d_g=O(1), we now claim that (<ref>) continues to hold with probability 1-C_1N^-1.4. To see this we assume N is large so g has no roots in B(λ_i(C_N)+ρ/λ_i(C_N),(δ_N)^3/4) except at the center of this disk, which may have multiplicity d_g or less. Thus we have (δ_N)^3d_g/4≲ |g(z)| on z∈∂ B(λ_i(C_N)+ρ/λ_i(C_N),(δ_N)^3/4). Then we can apply the isotropic law
in Theorem <ref> with the version where we have (δ_N)^d_g on the right hand side of (<ref>) and thus complete the proof.





 §.§ The non-Gaussian case
 The proof of the master theorem <ref> in the non-Gaussian case is a simple adaptation of the proof in the Gaussian case.

 We follow exactly the steps in the proof of Theorem <ref>, where we have the same free probability object 𝒴_z,free having the same Stieltjes transform. It suffices to use Theorem <ref>, <ref> and <ref> combined with (or in place of) Theorem <ref>, <ref> <ref> and all other steps are not changed: the former group of theorems determines concentration for the non-Gaussian model and compares the non-Gaussian model to the associated Gaussian model G or the free model 𝒴_z,free; whereas the latter group of theorems further compares the Gaussian model to the free model.





§ PROOF OF APPLICATIONS





 §.§ Truncation step


We can now prove Theorem <ref> and <ref> via a truncation argument.

 Let X:=(d_N)^-1/2X_N where X_N is specified in Definition <ref>. By definition we verify that 
    𝔼[XX^*]=𝔼[X^*X]=1
 and if all the self-loops (x,x)∈ E_N we have
    𝔼[X^2]=D/d_N1
   (recall 𝔼[g_ii^2]=D) where (log N)D/d_N→ 0 by assumption of d_N. Otherwise, if none of the self-loops (x,x)∈ E_N then simply  𝔼[X^2]=0.

    Therefore the claim of Theorem <ref> for Gaussian entries (case (1)) follow from Theorem <ref> with ρ=0, and the claim of Theorem <ref> for bounded entries (case (2)) follow from Theorem <ref> with ρ=0. As we assume the graphs G_N are increasing and X_N can be embedded as a submatrix of X_N+1, the almost sure statement follows from an application of Borel-Cantelli lemma.

It remains to prove Theorem <ref>  case (3), where entries are i.i.d. with finite p-th moment. We begin with a moment estimate similar to <cit.>, Corollary 3.32: 

    𝔼[sup_(x,y)∈ E_N|(d_N)^-1/2g_xy|^2]≤1/d_N𝔼[max_(x,y)∈ E_N|g_xy|^p]^2/p≤(Nd_N)^2/p/d_N𝔼[|g_xy^p]^2/p.
 By our assumption on d_N we have (d_N)^-1(Nd_N)^2/p<< (log N)^-4. Therefore we conclude that we can find a deterministic sequence a_N↘ 0 such that with probability 1-o(1),

    (log N)^2sup_(x,y)∈ E_N(d_N)^-1/2|g_xy|≤  a_N.
 For regularity reasons we also assume that a_N≥ (log N)^-4/p-2.


   Now we define a truncated matrix X_N on G_N such that for each edge (x,y)∈ E_N, the entry (X_N)_(x,y) is g_xy1_|g_xy|≤ a_N(d_N)^1/2(log N)^-2. We let 
   
    g_xy:=g_xy1_|g_xy|≤ a_N(d_N)^1/2(log N)^-2,
 since g_xy is symmetric we have 𝔼[g_xy]=0. We also compute that  V_N:=𝔼[|g_xy|^2]=1+O(N^-1) . This is because by assumption on d_N we have a_N(d_N)^1/2(log N)^-2≥ N^1/p-2, so that 
   
    𝔼[g_xy^2]-𝔼[g_xy^2]=𝔼[g_xy^21_|g_xy|≥ a_N(d_N)^1/2(log N)^-2]≤ N^-1𝔼[g_xy^p].
 
   
   
   Therefore we may instead consider Y:=(d_NV_N)^-1/2X_N and now this matrix satisfies the assumptions of Theorem <ref>, case (2) (we verify that the entries are bounded and 𝔼[YY^*]=𝔼[Y^*Y]=1, 𝔼[Y^2]=ρ(1+o((log N)^-1))1), so the claim on absence of outliers (part (1)) and on convergence of outlying eigenvalues (part (2)) hold for Y. Moreover, X_N=X_N on the event where (<ref>) holds (which has probability 1-o(1)) and V_N=1+o(N^-1). Then an elementary exercise enables us to transfer the finite rank perturbation results on Y to that of X. (To be more specific, we determine outlying eigenvalues of (d_N)^-1/2V_NX_N+V_NC_N, but the eigenvalues of the latter matrix are eigenvalues of (d_N)^-1/2X_N+C_N multiplied by V_N; and eigenvalues of V_NC_N are eigenvalues of C_N multiplied by V_N. Finally use V_N=1+O(N^-1).)

   To upgrade to an almost sure argument, we can use <cit.>, Lemma 9.22 and the fact that 𝔼[|g_xy|^p]<∞ to deduce that 
   
    lim_N→∞1/(Nd_N)^1/psup_(x,y)∈ E_N|g_xy|=0
          a.s.,
 then using our assumptions on d_N and <cit.>, Lemma 9.21, we deduce that we can find deterministic sequence a_N↘ 0, such that
   
    (d_N)^-1/2sup_(x,y)∈ E_N(d_N)^-1/2|g_xy|≤ (log N)^-2a_N eventually  a.s..
 Then we can upgrade the previous proofs to an almost sure statement.


For banded elliptic random matrices, a very similar argument can be used to prove Theorem <ref>.

 
    Let X:=(d_N)^-1/2X_N. Then by definition we have 
    
    𝔼[XX^*]=𝔼[X^*X]=1,  𝔼[X^2]=(ρ+o((log N)^-1)1.



    
    In case (1) when entries are Gaussian, the claim is implied by Theorem <ref>. In case (2) when entries are bounded, the claim is implied by Theorem <ref>.

    It remains to consider case (3) when (g_0,g_1,g_2) have finite p-th moment. As in the previous proof, applying <cit.>, Lemma 9.22 to the following three families of random variables {(X_N)_(x,y),x< y,(x,y)∈ E_N}, {(X_N)_(x,y),y< x,(x,y)∈ E_N} and {(X_N)_(x,x),(x,x)∈ E_N} (each group consists of i.i.d. random variables) and using the fact that 𝔼[|g_i|^p]<∞ for i=0,1,2, we conclude that    
    lim_N→∞1/(Nd_N)^1/psup_(x,y)∈ E_Nmax(|(X_N)_(x,y)|,|(X_N)_(y,x)|)=0
          a.s.,
 then using our assumptions on d_N and <cit.>, Lemma 9.21, we deduce that we can find a deterministic sequence a_N↘ 0, such that
   
    (d_N)^-1/2sup_(x,y)∈ E_N(d_N)^-1/2max(|(X_N)_(x,y)|,|(X_N)_(y,x)|)≤ (log N)^-2a_N eventually  a.s..

 We again assume that a_N≥ (log N)^-4/p-2, as this can be achieved by replacing a_N by max(a_N, (log N)^-4/p-2).


 
   Now we define a truncated matrix X_N on G_N such that for each edge (x,y)∈ E_N, the entry (X_N)_(x,y) is (X_N)_(x,y)1_|(X_N)_(x,y)|≤ a_N(d_N)^1/2(log N)^-2. Then X_N=X_N eventually almost surely.

From the random variables (g_0,g_1,g_2) we define the truncated version (g_0,g_1,g_2) via setting
g_ilaw=g_i1_|g_i|≤ a_N(d_N)^1/2(log N)^-2 for each i=0,1,2. Since g_0,g_1,g_2 are symmetric, we see g_0,g_1,g_2 still has mean zero. As in the previous proof, applying Markov's inequality, the fact that g_1law=g_2, and the finite p-th moment of g_i, we check that 

    𝔼[|g_1|^2]=𝔼[|g_2|^2]=1+O(N^-1),  𝔼[|g_0|^2]=1+O(N^-1),

   and that 
    𝔼[g_1g_2]=ρ+O(N^-1).




   After a simple computation, we see that we can find a constant V_N=1+O(N^-1) such that Y:=V_NX_N satisfies all the assumptions in Theorem <ref>. Then we have proven no outlier and finite rank perturbation theorems (i.e. the first and second conclusions of Theorem <ref>) for the matrix Y. By definition of V_N, this implies the same limit of outlying eigenvalues for X_N. (To be more specific, we determine outlying eigenvalues of (d_N)^-1/2V_NX_N+V_NC_N, but the eigenvalues of the latter matrix are eigenvalues of (d_N)^-1/2X_N+C_N multiplied by V_N; and eigenvalues of V_NC_N are eigenvalues of C_N multiplied by V_N. Finally use V_N=1+O(N^-1).) Since X_N=X_N eventually almost surely, this completes the proof for (d_N)^-1/2X_N.






 §.§ product elliptic matrices


Finally we give the proof of Theorem <ref>. The proof relies on a standard linearization argument.






We consider the following two linearization matrices

    𝒳_N:=[                     0     (d_N,1)^-1/2X_N^1                                                                 0;                     0                     0     (d_N,2)^-1/2X_N^2                                           0;                                                                 ⋱                     ⋱                      ;                     0                                                                 0 (d_N,m-1)^-1/2X_N^m-1;     (d_N,m)^-1/2X_N^m                                                                                       0 ].

and the mN× mN matrix

    𝒜_N:=[       0   A_N^1       0;       0       0   A_N^2       0;               ⋱       ⋱;       0       0 A_N^m-1;   A_N^m       0 ].



From elementary linear algebra (see <cit.>, Proposition 4.1), we readily see that if λ is an eigenvalue of 𝒳_N, then λ^m is an eigenvalue of (𝒳_N)^m, so that λ^m is an eigenvalue of D_N^m. Similarly, if λ is an eigenvalue of 𝒜_N (or 𝒳_N+𝒜_N), then λ^m is an eigenvalue of A_N (or D_N^m,1).

Let ζ_m:={z∈ℂ:z^m=1} denote the set of m-th roots of unity. Then the eigenvalues of 𝒜_N are given by 
    ({λ_i(A_N))^1/mζ_m,   i=1,⋯,j},
 where we use ({λ_i(A_N))^1/m to denote any m-th square root of λ_1(A_N).

For the matrix 𝒳_N, we readily check that whenever m≥ 2, 
    𝔼[𝒳_N(𝒳_N)^*]=𝔼[(𝒳_N)^*𝒳_N)]=1,  𝔼[(𝒳_N)^2]=0.
 

Then we can apply Theorem <ref> and <ref> to 𝒳_N+𝒜_N (when all entries of 𝒳_N are Gaussian in case(1) or bounded in case (2)), and conclude that for any ϵ>0 a.s. for N large there are no eigenvalues of 𝒳_N outside ℂ∖ (1+ϵ)^1/m𝔻, so that  there are no eigenvalues of D_N^m outside (1+ϵ)𝔻. Moreover, almost surely as N is large, there are precisely mj outlying eigenvalues of 𝒳_N+𝒜_N in ℂ∖ (1+ϵ)^1/m𝔻, and after relabeling they converge to (λ_i(A_N))^1/mζ_m(1+o(1)) for each i=1,⋯,j. This implies that almost surely for N large, there are precisely j outlying eigenvalues of D_N^m,1 in ℂ∖ (1+ϵ)𝔻, and after relabeling, these eigenvalues converge to (λ_i(A_N))(1+o(1)) for each i=1,⋯,j. 

Finally, when entries of 𝒳_N have a bounded p-th moment, we may use a similar truncation argument as in the proof of Theorem <ref>, <ref>. The adaptations are straightforward and hence omitted.
    







§ PROOF OF CONCENTRATION INEQUALITIES


In this appendix we outline a proof of Theorem <ref> and <ref>: the proof is essentially an adaptation of the version in <cit.>. 



We write the Gaussian model G as

    G:=A_0+∑_i=1^n g_iA_i
 where g_1,⋯,g_n are independent mean 0, variance 1 Gaussians and A_i∈ M_n(ℂ)_sa are fixed self-adjoint matrices. Define f:ℝ^n→ℝ as

    f(x):=⟨ u,(z1-A_0-∑_i=1^nx_iA_i)^-1v⟩.

We can verify as in <cit.>, Lemma 6.5 that f is σ_*(G)/(Im z)^2 Lipschitz continuous, so that ⟨ u,(z1-X)^-1v⟩ is a σ_*(G)/(Im z)^2 Lipschitz continuous function of n unit Gaussian variables. Then the Gaussian concentration inequality from <cit.> Theorem 5.6 finishes the proof.
    



 We will adapt the proof of <cit.> and only give a sketch. Some estimates we present are suboptimal, but we will use them as we can simply quote the computations from the cited work. 


Let X be of the form <ref> and consider (Z_1',⋯,Z_n') to be some independent copy of (Z_1,⋯,Z_n). Let X^∼ i:=Z_0+∑_j≠ iZ_j+Z_i'. Then for any fixed unit vectors u,v∈ℝ^N, 

    ∑_i=1^n (⟨ u,(z1-X)^-1v-(z1-X^∼ i)^-1v⟩)_+^2≤ W,
where we define W via

    W:=2/(Imz)^4sup_v=w=1∑_i=1^n|⟨ v,(Z_i-Z_i')w⟩|^2+8/(Imz)^6R(X)^2∑_i=1^n(Z_i-Z_i')^2.



Then we can estimate as in Lemma 5.8 of <cit.>,

    𝔼[W]≲σ_*(X)^2/(Imz)^4 +R(X)𝔼X-𝔼X/(Imz)^4+R(X)^2𝔼X-𝔼X^2/(Imz)^6.
 W has the following self-bounding property: consider

    W^∼ i:=2/(Imz)^4sup_v=w=1∑_j≠ i|⟨ v,(Z_j-Z_j')w⟩|^2+8/(Imz)^6R(X)^2∑_j≠ i^n(Z_j-Z_j')^2,

    then by <cit.>, Lemma 5.9, W^∼ i≤ W and 
    
    ∑_i=1^n(W-W^∼ i)^2≤{16R(X)^2/(Imz)^4+64R(X)^4/(Imz)^6}W.


    From the self-bounding property of W we get 
    log𝔼[e^W/a]≤2/a𝔼[W],   a=16R(X)^2/(Imz)^4+64R(X)^4/(Imz)^6.

    Then for 0≤λ≤ a^-1/2, by exponential Poincaré inequality and Chernoff bound, we have

    ℙ[⟨ u,(z1-X)^-1v⟩≥𝔼⟨ u,(z1-X)^-1v⟩+√(8𝔼Wx)+√(a)x
    ]≤ e^-x.


We can also prove via a refinement of <cit.> Theorem 6.16 that for any 0≤λ≤ (2a)^-1/2,

    log𝔼[e^-λ{⟨ u,(z1-X)^-1v⟩-𝔼⟨ u,(z1-X)^-1v⟩}]≤4𝔼[W]λ^2/1-λ√(2a).
Then Chernoff's inequality imply the lower tail bound for any x>0

    ℙ[⟨ u,(z1-X)^-1v⟩≤𝔼⟨ u,(z1-X)^-1v⟩-4√(𝔼Wx)-√(2a)x
    ]≤ e^-x.

This completes the proof with both upper and lower tails derived.
    




§ STATEMENTS AND DECLARATIONS

The author has no financial or non-financial conflicting of interests to declare that are relevant to the contents of this article.

No dataset is generated in this research.









 Paper Title: Large deviations for the maximum of a reducible two-type branching Brownian motion.







§ ABSTRACT 
We consider a two-type reducible branching Brownian motion, defined as a particle
system on the real line in which particles of two types move according to independent
Brownian motions and create offspring at a constant rate. Particles of type 1 can give
birth to particles of types 1 and 2, but particles of type 2 only give birth to descendants
of type 2. Under some specific conditions, Belloum and Mallein in <cit.> show that  the maximum position M_t of all particles alive at time t, suitably centred  by a deterministic function m_t, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as t→∞,

    (M_t≥θ m_t),  θ>1.

We shall show that  the decay rate function exhibits  phase transitions depending on  certain relations between θ, the variance of the underling Brownian motion and the branching rate. 


[2010]60F10, 60J80



MUFASA: Multi-View Fusion and Adaptation Network with Spatial Awareness for Radar Object Detection
    Xiangyuan Peng1,2 Miao Tang3 Huawei Sun1,2Kay Bierzynski2Lorenzo Servadei1 Robert Wille1

    August 5, 2024
==================================================================================================





§ INTRODUCTION AND MAIN RESULTS



Branching Brownian motion (BBM) is a spatial branching process of great interests in recent years. On the one hand, its connection with the partial differential Fisher-Kolmogorov-Petrovsky-Piskunov (F-KPP) equation brings out many studies on both probabilistic and analytic sides, see for instance, <cit.>. On the other hand, it is the fundamental model to understand the BBM-universality class which includes the 2-dim Gaussian free field <cit.> and 2-dim cover times <cit.>, etc.




An one-dimensional  standard binary branching Brownian motion is  a continuous-time particle system on the real line which can be constructed as follows. It starts with one individual located at the origin at time 0 that moves according to a Brownian motion with variance σ^2>0. After an independent exponential time of parameter β, the initial particle dies and gives birth to 2 children that start at the position their parent occupied at its death. These 2 children then move according to independent Brownian motions with variance σ^2 and give birth independently to their own children at rate β. The particle system keeps evolving in this fashion for all time.



In this work, we shall study the two-type reducible branching Brownian motion. This is a particle system on the real line in which particles possess a type in addition with their position. Particles of type 1 move according to Brownian motions with variance σ^2 and branch at rate β into two children of type 1. Additionally, they give birth to particles of type 2 at rate α. Particles of type 2 move according to Brownian motions with variance σ_2^2 and branch at rate β_2, but cannot give birth to descendants of type 1. Up to a dilation of time and space, in this work we shall always assume  that β_2=σ_2=1.  This model have been studied extensively by many authors; see <cit.>.




With abuse of notation, for all t ≥ 0, we denote by _t the collection of the individuals alive at time t for both the standard BBM and the two-type reducible BBM. And denote by _t^1 and _t^2 the set of particles of type 1 and type 2 at time t, respectively.  For any u∈(t) and s ≤ t, let X^u_s denote the position at time s of the individual u or its ancestor alive at that time. The maximum of the branching Brownian motion at time t is defined as

    M_t:=max{X^u_t: u∈(t)}.


To distinguish the two-type reducible BBM and the standard BBM,  we shall regard the standard  BBM as (σ^2,β)-BBM to emphasis the variance of the Brownian motion  and the branching rate. We also  denote by _σ^2, β its law and write X̅^u_s for the position at time s of the individual u of a (σ^2, β)-BBM.

For (σ^2, β)-BBM,  the asymptotic behaviour of

    M̅_t:=max{X̅_t^u: u∈(t)},

as t →∞, has been subjected to intense study, partly due to its link with the F-KPP reaction-diffusion equation, defined as

    ∂_t u(t, x)=1/2∂^2_xu(t, x)+(1-u(t,x))-(1-u(t,x))^2,   t≥0,  x∈ℝ.

Precisely, McKean <cit.> showed that 

    _σ^2, 1(M̅_t ≥σ x)=u̅(t,  x),
 
where the function (t,x) ↦u̅(t, x) is the unique solution of (<ref>) with initial condition u̅(0, x) = _{x < 0}. With the help of (<ref>), Bramson <cit.> proved that for a (σ^2, β)-BBM,

    M̅_t-√(2 σ^2 β) t+3/2 √(2 β / σ^2)ln t,

converges weakly, as t→∞. In particular, uniformly in x ∈, as t→∞,

    1-u̅(t, m̅_t+x)=_σ^2, 1(M̅_t/σ-√(2)t+3/2√(2)ln t≤ x )⟶ω̅(x),

where ω̅ is the travelling wave solution of the F-KPP equation at speed √(2), which satisfies

    1/2ω̅” + √(2)ω̅' -  ω̅(1 -  ω̅) = 0.

 Lalley and Sellke <cit.>  characterised  ω̅ in terms of  a random shift of the Gumbel distribution by introducing the so-called derivative martingale.

Parallel results of (<ref>) for the two-type reducible BBM were obtained in <cit.>. To state their results, we need to decompose the state space (β, σ^2)∈ℝ_+^2 into three regions:


    𝒞_1    ={(β, σ^2): σ^2>1_{β≤ 1}/β+1_{β>1}β/2 β-1}, 
    𝒞_2    ={(β, σ^2): σ^2<1_{β≤ 1}/β+1_{β>1}(2-β)}, 
    𝒞_3    ={(β, σ^2): σ^2+β>2  and σ^2<β/2 β-1} .


From Belloum and Mallein <cit.>, we have:

    M_t-m_t

converges weakly, as t→∞, where

    m_t=√(2 σ^2 β) t-3/2 √(2 β / σ^2)ln t,      (β, σ^2)∈𝒞_1,√(2) t-3/2 √(2)ln t,      (β, σ^2)∈𝒞_2,σ^2-β/√(2(1-σ^2)(β-1)) t,     (β, σ^2)∈𝒞_3.

From above, we see that if (β, σ^2) ∈𝒞_I, the speed of the two-type reducible BBM is √(2 βσ^2), which is the same as particle s of type 1 alone, ignoring births of particles of type 2.  Conversely, if (β, σ^2) ∈𝒞_2, then the speed of the process is √(2), equal to the speed of a single BBM of particles of type 2.  Finally, if (β, σ^2) ∈𝒞_3, the speed of the process is a mixture of the long-time asymptotic of the processes of particles of type 1 and 2,  which is larger than max(√(2), √(2 βσ^2)).

 In this article, we study the large deviation probabilities for the maximum of the two-type reducible BBM. That is the convergence rate of the following probability 
 
    (M_t> θ m_t),  θ>1,

as t→∞. We divide the main result of our article into three theorems, one for each area the pair (β, σ^2) belongs to.  










If (β, σ^2) ∈𝒞_1, then
the following limit exists for any θ>1,


    lim _ t→∞1/tln(M_t≥√(2 βσ^2 )θ t )=A,



where for β <1,


    A=β(1-θ^2),


for β >1 and  σ ^2≥ 1, we have


    A=β(1-θ^2),


for β >1 and σ ^2< 1, we have


    A=β(1-θ^2),    θ^2≤(β -1)σ ^2/(1-σ^2)  β, 
    
    1-θ^2 βσ^2,    θ^2≥(β-1) /(1-σ^2) σ ^2β ,
    -2 √(β(β-1)(1-σ^2))σθ+β-σ^2/1-σ^2,    (β -1)σ ^2/(1-σ^2)  β≤θ^2≤(β-1) /(1-σ^2) σ ^2β.






  If (β, σ^2) ∈𝒞_2, then the following limits exists for any θ>1,


  
    lim _ t→∞1/tln(M_t≥√(2)θ t )=:A,



  where for β >1, we have

  
    A=1-θ^2 ,


  for β≤ 1, σ ^2 ≤   1, we have

  
    A=1-θ^2,


  for β≤ 1, σ ^2 > 1, we have

  
    A=1- θ^2,    θ^2≤(β -1) /(1-σ^2) ,
      β -θ ^2/σ ^2,   θ^2≥(1-β) /(σ^2-1) σ ^4, 
        2 √((β-1)(1-σ^2))θ+β-σ^2/1-σ^2,    (β -1) /(1-σ^2)  <  θ^2 <  (1-β) /(σ^2-1) σ ^4.

  


 If (β, σ^2) ∈𝒞_3, then for any θ>1, we have
 
    lim _ t→∞1/tln(M_t≥ξθ t )=
    1-ξ^2θ^2/2,    θ^2≥2(β-1)/1-σ^21/ξ^2,
    β-ξ^2θ^2/2 σ^2,    θ^2≤2(β-1)/1-σ^2σ^4/ξ^2,
      √(2)(1-β)-θ^2(σ^2-β)^2-1/√(2)(σ^2-1)+1,    2(β-1)/1-σ^2σ^4/ξ^2 < θ^2< 2(β-1)/1-σ^21/ξ^2,

where ξ =σ ^2-β/√(2(1-σ ^2)(β -1)).


Let us mention here that in the context of standard (1,1)-BBM (i.e., σ^2=1, β=1), one would have: For any θ>1 and x∈ℝ,
 
    lim_t→∞√(t)e^(θ^2-1)t_1,1(M̅_t≥√(2)θ t+x)   
     =lim_t→∞√(t)e^(θ^2-1)tu̅(t, √(2)θ t+x)
         =e^√(2)xC(θ, u̅),

where C(θ, u̅) is a constant depending only on θ and u̅; see <cit.>, <cit.> and <cit.>. In this work, we only obtain the exponential decay rates for the large deviation probabilities for the maximum of two-type inreduciable BBM. It is possible to obtain more preciser form as (<ref>) with the help of truncated second moment method and the F-KPP equation. Our method may also apply to study the cases of (β, σ^2)∈𝒞_1∩𝒞_2 or (β, σ^2)∈𝒞_2∩𝒞_3 or  (β, σ^2)∈𝒞_1∩𝒞_3; see <cit.> for related results.


In the next section, we will present some technical lemmas borrowed from existed literatures.  The proofs for Theorems <ref>, <ref> and <ref> will be given in Sections <ref>, <ref> and <ref>, respectively.



§ TECHNICAL LEMMAS

(<cit.>, Lemma 2.1)
Let X be a centered Gaussian random variables with variance σ^2>0. Then

    (X>a)=(2π)^-1/2(σ/a)
    e^-a^2/2σ^2(1+O(σ^2/a^2)),  as  a/σ→∞,

with the r.h.s. above without error term being an upper bound valid for any a>0.



 
Furthermore,  for (σ^2, β)-BBM and 0<x<1, _σ^2,β-a.s.,

    1/tlog_t(√(2β)xσ t))t→∞⟶ (1-x^2)β,

where  _t(x) denote the number of particles, in the (σ^2, β)-BBM, alive at time t and positioned in [tx, ∞); see <cit.>.

 If u ∈𝒩_t^2, we denote by T(u) the time at which the oldest ancestor of type 2 of u was born. The set 𝔅 of particles of type 2 that are
born from a particle of type 1, that can be defined as 

    𝔅:={u∈∪_t≥ 0^2_t: T(u)=b_u},

where b_u is the birth time of u.
We observe that 𝔅 can be thought of as a Poisson point process with random intensity.
(<cit.>, Lemma 4.2)
Set ℱ_1 =σ(X_t^u, u∈_t^1,t≥0). The point measure ∑_u∈𝔅δ_(X_s^u, s≤ T(u))
conditionally on ℱ_1  is a Poisson point process with intensity α dt ×∑_u∈𝒩_t^1δ_(X_s^u, s≤ t).

The results in the following proposition are variations of the so-called many-to-one lemma.
  For all measurable non-negative functionals f and g, we have

    [∑_u∈_t^1 f(X_s^u, s ≤  t)]
    =
     e^β t[f(σ B_s , s≤ t)]

and

    [∑_u∈_t^2 g((X_s^u, s ≤  t), T(u))]      =
    α∫_0^t e^β s+(t-s)E[g((σ B_u∧ s +(B_u-B_u∧ s), u≤ t),s)] d s.



The first result is just the usual many-to-one lemma. The second one can be found in Proposition 4.1 in <cit.>. 



§ PROOF OF THEOREM <REF>

In this section, we consider the case that

    (β, σ^2)∈𝒞_1 ={(β, σ^2): σ^2>1_{β≤ 1}/β+ β1_{β>1}/2 β-1}.



 §.§ Upper bound


From Lemma <ref>, we have


    (M_t ⩾√(2 σ^2 β)θ t) 
    =    (∃  u ∈_t, X_t^u ⩾√(2 σ^2 β)θ t) ⩽   (∃  u ∈_t^1, X_t^u ⩾√(2 σ^2 β)θ t)+(∃  u ∈_t^2, X_t^u ⩾√(2 σ^2 β)θ t) ⩽    2max[ e^β t(σ B_t ⩾√(2 σ^2 β)θ t),α∫_0^t e^β s+(t-s)(σ B_s+B_t-B_s ⩾√(2 σ^2 β)θ t) ṣ]  
     =:    2max{ I_1, I_2}.


Then by (<ref>)

    I_1    ≤ O(1) ·1/√(t)· e^β t· e^-2 βθ^2 t^2/2 t
        =O(1)/√(t)· e^β(1-θ^2) t.



Noticing that

    Var(σ B_s+B_t-B_s)=σ^2 · s+(t-s),


we have

    I_2    ≤α∫_0^t e^β s+(t-s)·1/√(2 π(σ^2 s+t-s))· e^-2 σ^2 βθ^2 t^2/2(σ^2 s+t-s)ṣ    = α∫_0^1 e^f(u)t·√(t)/√(2 π((σ^2 -1)u+1))ụ,
 
where 

    f_1(u)=(β-1) u+1-σ^2 βθ^2/(.σ^2-1) u+1.


Notice that f_1(1)=β(1-θ^2). Then our task is reduced to know max_0≤ u≤ 1f(u). 
To end this, let us consider

    f_1^'(u)=β-1+σ^2 βθ^2/[(σ^2-1) u+1]^2(σ^2-1).



We have three cases.

i). If (β, σ^2) ∈𝒞_1 and β≤ 1, then
u↦ f_1^'(u) is an increasing function. In this case, by noticing that βσ^2 >1, one has

    f_1^'(0)
     =β-1+σ^2β (σ^2-1)
     =β(1-σ^2)+σ^2βσ^2-1
     > β(1-σ^2)+σ^2-1
     ≥ 0·

 Thus f_1'(u)≥ 0 for all u∈ [0,1]. Hence max_0≤ u≤ 1f_1(u)=f_1(1) and then we have

    lim sup_t →∞1/tln(M_t>√(2 σ^2 β)θ t)≤β(1-θ^2).


ii). If (β, σ^2) ∈𝒞_1 and β≥ 1,σ ^2 ≥ 1, then 
f_1^'(u)≥ 0.  Thus max_0≤ u≤ 1f_1(u)=f_1(1) and hence

    lim sup_t →∞1/tln(M_t>√(2 σ^2 β)θ t)≤β(1-θ^2).


iii). If (β, σ^2) ∈𝒞_1 and β >1,σ ^2 <  1, then obviously u↦ f_1^'(u) is decreasing. Then we have three subcases.

First, if θ^2<  (β -1)σ ^2/(1-σ^2)  β, which implies f^'(1)>0, then max_0≤ u≤ 1f_1(u)=f_1(1) and 

    lim sup_t →∞1/tln(M_t>√(2 σ^2 β)θ t)≤β(1-θ^2).


Second, if  θ^2> (β-1) /(1-σ^2) σ ^2β,  which means f_1^'(0)<0,  then max_0≤ u≤ 1f_1(u)=f_1(0) and 

    lim sup_t →∞1/tln(M_t>√(2 σ^2 β)θ t)≤  1-θ^2 βσ^2.


Last, if (β -1)σ ^2/(1-σ^2)  β≤θ^2≤(β-1) /(1-σ^2) σ ^2β, then f_1^'(0)≥ 0 and f^'(1)≤ 0. Thus max_0≤ u≤ 1f_1(u)=f_1(u_0), where f_1^'(u_0)=0. And one can check that


    u_0=√(|σ^2-1/1-β|)·σθ·√(β)-1/|σ^2-1|,

 and 

    f_1(u_0)=-2 √(β(β-1)(1-σ^2))σθ+β-σ^2/1-σ^2.
 Therefore,

    lim sup_t →∞1/tln(M_t>√(2 σ^2 β)θ t)≤-2 √(β(β-1)(1-σ^2))σθ+β-σ^2/1-σ^2.

This proves the upper bound.





 §.§ Lower bound for Theorem <ref>


Notice that

    (M_t ⩾√(2 βθ^2 σ^2)·  t) 
    ⩾   max{(∃  u ∈_t^1: X_t^u ⩾√(2 βθ^2σ^2)·  t), (∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t)}.


First, from <ref>, we have

    lim_t→∞1/tln(∃  u ∈_t^1: X_t^u ⩾√(2 βθ^2σ^2)·   t)     =lim_t→∞1/tln_σ^2, β(M̅_t ⩾√(2 βθ^2 σ^2 )· t)    = β(1-θ^2).




On the other hand, set
    𝔅_t= {u∈𝔅: T(u)<t}.

According to construction of our model, we have

    (∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t)     =  (∃ u ∈𝔅_t : X_T(u)^u+M_t-T(u)^u ⩾√(2 βθ^2σ^2)·  t)    =  1-(∀  u ∈𝔅_t : X_T(u)^u+M_t-T(u)^u < √(2 βθ^2σ^2)·  t)    =  1-[∏_u ∈𝔅_t φ_t(X_T(u)^u, T(u))]    =  1-[exp(-∑_u ∈𝔅_t -lnφ_t(X_T(u)^u, T(u)))],



where 
    1- φ_t(x, s)  = u̅(t-s, √(2 βθ^2σ^2)·  t-x)
        = _1,1(M̅_t-s≥√(2 σ ^2θ^2 β)· t-x).
 
and  {M^u_t-T(u): u∈𝔅_t} are independent random variables such that M^u_t-s has the same law as M̅_t-s under _1, 1. Applying lemma <ref>, we have

    (∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t)     =   1-[exp{-α∫_0^t ∑_u∈_s^1u̅(t-s, √(2 βθ^2 σ^2)·  t-X_s^u) d s}]
         ≥ 
      1-[exp{-α∫_I_t∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-X_s^u) d s}]
           ≥ 
      1-[exp{-α∫_I_t∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t ) d s}]     ≥ 
      1-[exp{-α∫_I_t∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t ) d s}],

where I_t⊂ [0, t] is an interval, x_t∈ℝ is a deterministic function of t, and

    _s^1(a)={u∈_s^1: X_s^u≥ a}.

Both I_t and x_t will be determined later from case to case. Furthermore, by Jensen's inequality and Fubini's theorem, we have 


    [exp{-α∫_I_t∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t ) d s}]    ≤∫_I_t[exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t )}]  d s.







  §.§.§ The case of max_0≤ u≤ 1 f_1(u)=f_1(0). 


Choose 

    I_t=[ε t, ε t+1],   x_t=-ε t,

for  ε>0 small enough. Then for s∈ I_t,

    [exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t )}]    ≤[exp{-α∑_u∈_s^1(-ε t)u̅(t-s, √(2 βθ^2 σ^2)·  t+ε t )}, |_s^1(-ε t)  | ≥ 1]       + ( |_s^1(-ε t)  | =0  )
       ≤exp{-αu̅(t-s, √(2 βθ^2 σ^2)·  t+ε t )}_σ^2, β( M̅_s≥  -ε t) + _σ^2, β( M̅_s < -ε t).



Therefore,

    (∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t)    ≥ 
      1-
     ∫_I_t[exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t )}]  d s
        ≥∫_I_t(1- exp{-αu̅(t-s, √(2 βθ^2 σ^2)·  t+ε t )})_σ^2, β( M̅_s≥  -ε t)  d s
        ≥∫_0^1  (1- exp{-αu̅(t-ε t-s, √(2 βθ^2 σ^2)·  t+ε t )})_σ^2, β( M̅_ε t+s≥  -ε t)  d s.



Then applying (<ref>), we have 

    lim_ε→ 0lim_t →∞1/tlnu̅(t-s, √(2 βθ^2 σ^2)·  t+ε t )≥  1-σ^2 βθ^2,

uniformly for s∈ [0,1]. We further  notice  that 
 
    _σ^2, β( M̅_s≥ -ε t)
      → 1,  as t→∞,

uniformly for s∈ [0,1]. Then applying Jensen's inequality and Fatou's lemma, one has

    lim inf_ε→ 0lim inf_t →∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t) 
          =
      1-σ^2 βθ^2
         ≥β (1-θ^2),

where the last inequality holds if θ^2≥(β-1) /(1-σ^2) β.
This proves Theorem <ref> for the case of θ^2≥(β-1) /(1-σ^2) σ ^2β, when σ^2<1 and β>1.






  §.§.§ The case of max_0≤ u≤ 1 f_1(u)=f_1(1). 
 

This case corresponds to the case that

    {(θ^2, β, σ^2):θ^2>1, β≤ 1, σ^2>1/β}⋃{(θ^2, β, σ^2):θ^2>1, β>1, σ^2>1}     ⋃{(θ^2, β, σ^2): θ^2≤(β -1)σ ^2/(1-σ^2)  β, β>1, β/2β-1<σ^2<1}.




Choose 

    I_t=[(1-ε) t, (1-ε) t+1],   x_t=√(2 βσ^2 )θ t+ε t,

for  ε>0 small enough. Then for s∈ I_t,

    [exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t )}]    ≤[exp{-α∑_u∈_s^1(x_t)u̅(t-s, -ε t )}, |_s^1(x_t)  | ≥ 1]       + ( |_s^1(x_t)| =0  ).



Notice that

    ( |_s^1(x_t)| ≥ 1  )= _σ^2, β( M̅_s> x_t  ).


Therefore,

    (∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t) 
        ≥∫_I_t( 1- exp{-α∑_u∈_s^1(x_t)u̅(t-s, -ε t )}|_s^1(x_t)  | ≥ 1 )_σ^2, β( M̅_s> x_t  )  d s
        ≥∫_I_t( 1- exp{-αu̅(t-s, -ε t )}|_s^1(x_t)  | ≥ 1 )_σ^2, β( M̅_s> x_t  )  d s
        =
    ∫_0^1  ( 1- exp{-αu̅(ε t-s, -ε t )}|_(1-ε) t+s ^1(x_t)  | ≥ 1 )_σ^2, β( M̅_(1-ε) t+s > x_t  )  d s.


Since 

    lim_t→∞u̅(ε t-s, -ε t )= 1

and

    lim_ε→0lim inf_t→∞1/tln_σ^2, β( M̅_(1-ε) t+s > x_t  )
     =β (1-θ^2),

uniformly for s∈ [0,1], 
then we have

    lim inf_t→∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t)=β (1-θ^2).









  §.§.§ The case of max_0≤ u≤ 1 f_1(u)=f_1(u_0). 


Notice that where f_1'(u_0)=0.  This  corresponds to  the case that
β >1, σ ^2< 1, and 


    (β -1)σ ^2/(1-σ^2)  β≤θ^2≤(β-1) /(1-σ^2) σ ^2β.

We also recall that

    f(u_0)=-2 √(β(β-1)(1-σ^2))σθ+β-σ^2/1-σ^2.




For  ε>0 small enough, choose 

    I_t=[u_0 t, u_0 t+1],   x_t=x_0 t,

for some 0<x_0< √(2 βθ^2 σ^2), whose value will be determined later.  Then for s∈ I_t,

    [exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t )}]    ≤[exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_0 t )}, |_s^1(x_t)  | ≥ e^(β_0-ε) t]       + _σ^2,β( |_s^1(x_t)  | < e^(β_0-ε) t)   ≤exp{-α   e^(β_0-ε) tu̅(t-s, √(2 βθ^2 σ^2)·  t-x_0 t )}( |_s^1(x_t)  | ≥ e^(β_0-ε) t)       + ( |_s^1(x_t)  | < e^(β_0-ε) t),

where 

    β_0=  β u_0 -x_0 ^2/2 σ^2 u_0>0,  and   
    x_0=√(2β)θ u_0 σ^3/1-u_0+u_0 σ ^2.

Therefore,

    (∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t) 
        ≥∫_0^1  (1-exp{-α   e^(β_0-ε) tu̅(t-u_0t-s, √(2 βθ^2 σ^2)·  t-x_0 t )})          ×_σ^2,β( |_u_0t+s^1(x_t)  | ≥ e^(β_0-ε) t)  d s.






We also notice that

    (√(2βσ ^2)θ-x_0)^2/2(1-u_0)^2>1,

since

    σ^2≥β/2β-1>2-β,

for β>1. Then applying (<ref>) again, we have 

    lim inf_t →∞1/tlnu̅(t-u_0 t-s, √(2 βθ^2 σ^2)·  t-x_0 t )
         ≥(1-(√(2βσ ^2)θ-x_0)^2/2(1-u_0)^2)(1-u_0),

uniform for s∈ [0,1].  Furthermore, according to (<ref>),

    1/tlog |_u_0 t^1(x_t) |→β_0,    a.s.

Thus

    lim_t→∞( |_u_0 t+s^1(x_t)  | ≥ e^(β_0-ε) t) = 1.


Since ε>0 is arbitrary, then applying Jensen's inequality and Fatou's lemma, one has

    lim inf_t →∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2 βθ^2σ^2)·   t) 
        ≥β_0+(1-(√(2βσ ^2)θ-x_0)^2/2(1-u_0)^2)(1-u_0)
         = (β-1) u_0+1-σ^2 βθ^2/(.σ^2-1) u_0+1     = 
    -2 √(β(β-1)(1-σ^2))σθ+β-σ^2/1-σ^2   ≥β(1-θ^2)=f_1(1),

where the last inequality holds for the case that
β >1, σ ^2< 1, and 

    (β -1)σ ^2/(1-σ^2)  β≤θ^2≤(β-1) /(1-σ^2) σ ^2β.


This proves Theorem <ref> for the above case. 





















§ PROOF OF THEOREM <REF>

In this section, we always suppose that

    (β, σ^2)∈𝒞_2  ={(β, σ^2): σ^2<1_{β≤ 1}/β+1_{β>1}(2-β)}.

The arguments are exactly the same to those in Section <ref>. One just to replace √(2βσ^2)θ by √(2)θ. We only give an outline here. 




 §.§ Upper bound for Theorem <ref>

We first consider the upper bound.

First,

    (M_t ⩾√(2 )θ t)    ⩽ 2max[ e^β t(σ B_t ⩾√(2 )θ t),α∫_0^t e^β s+(t-s)(σ B_s+B_t-B_s ⩾√(2 )θ t) d s]   =:2max ( 1, 2).



By (<ref>), we have 

    1≤ O(1) ·1/√(t)· e^β t· e^-2  θ^2 t^2/2σ ^2 t =O(1)/√(t)· e^(β -θ^2/σ ^2) t.

and

    2≤α∫_0^t e^β s+(t-s)·1/√(2 π(σ^2 s+t-s))· e^-2 θ^2 t^2/2(σ^2 s+t-s).

Set


    f_2(u)=(β-1) u+1-θ^2/(.σ^2-1) u+1,    u ∈(0,1).


Then

    f_2^'(u)=β-1+θ^2/[(σ^2-1) u+1]^2(σ^2-1).

Solving f^'(u_0)=0, we have


    u_0=√(|σ^2-1/1-β|)·θ  -1/|σ^2-1|,


and 

    f_2(u_0)=2 √((β-1)(1-σ^2))θ+β-σ^2/1-σ^2.


Then we have several subcases.

Case I. (β, σ^2) ∈𝒞_2 and β >1. 


In this case u↦ f_2^'(u) is increasing and f_2^'(1)< 0. Thus max_0≤ u≤ 1f_2(u)=f_2(0) and hence

    lim _t →∞1/tln(M_t>√(2 )θ t)≤  1-θ^2.


Case II. (β, σ^2) ∈𝒞_2 and β≤ 1,σ ^2 < 1.



In this case f_2^'(u)< 0 for all u∈ [0,1], Thus max_0≤ u≤ 1f_2(u)=f_2(0) and hence


    lim _t →∞1/tln(M_t>√(2 )θ t)≤  1-θ^2.


Case III. (β, σ^2) ∈𝒞_2 and β≤ 1,σ ^2 ≥   1. 
In this case, u↦ f_2^'(u) is decreasing.

When  θ^2<(β -1) /(1-σ^2), then f_2^'(0)< 0. Thus max_0≤ u≤ 1f_2(u)=f_2(0) and hence

    lim _t →∞1/tln(M_t>√(2 )θ t)≤  1- θ^2.


When θ^2>(1-β) /(σ^2-1) σ ^4, then f_2^'(1)>0, Thus max_0≤ u≤ 1f_2(u)=f_2(1) and hence

    lim _t →∞1/tln(M_t>√(2 )θ t)≤β -θ ^2/σ ^2.


When (β -1) /(1-σ^2) ≤θ^2 ≤(1-β) /(σ^2-1) σ ^4, then f_2^'(0)≥ 0 and f_2^'(1)≤ 0. Thus max_0≤ u≤ 1f_2(u)=f_2(u_0) and hence

    lim _t →∞1/tln(M_t>√(2 )θ t)≤2 √((β-1)(1-σ^2))θ+β-σ^2/1-σ^2.







 §.§ Lower bound for Theorem <ref>


Notice that

    (M_t ⩾√(2)θ  t) 
    ⩾   max{(∃  u ∈_t^1: X_t^u ⩾√(2)θ  t), (∃  u ∈_t^2: X_t^u ⩾√(2)θ   t)}.


First, from <ref>, we have

    lim_t→∞1/tln(∃  u ∈_t^1: X_t^u ⩾√(2 )θ   t ) 
        =lim_t→∞1/tln_σ^2, β(M̅_t ⩾√(2 )θ t)     = β-θ^2/σ^2.


By similar reasoning as (<ref>), (<ref>) and (<ref>), we have

    (∃  u ∈_t^2: X_t^u ⩾√(2)θ   t)
           ≥∫_I_t1-   [exp{-α∑_u∈_s^1(x_t)u̅(t-s, √(2 βθ^2 σ^2)·  t-x_t )}]  d s,

where I_t⊂ [0, t] is an interval, x_t∈ℝ is a deterministic function of t. 
Both I_t and x_t will be determined later from case to case. Recall from (<ref>) that
 
    f_2(u)=(β-1) u+1-θ^2/(.σ^2-1) u+1,    u ∈(0,1).





  §.§.§ The case of max_0≤ u≤ 1 f_2(u)=f_2(0). 



This corresponds to the cases that 
 β >1, or β≤ 1, σ ^2 ≤   1, or β≤ 1, σ ^2 > 1 and θ^2<(β -1) /(1-σ^2) . The arguments are parallel to those in Section <ref>. We only give an outline. Choose 

    I_t=[ε t, ε t+1],   x_t=-ε t,

for  ε>0 small enough. Then 

    (∃  u ∈_t^2: X_t^u ⩾√(2 )θ   t)    ≥∫_0^1  (1- exp{-αu̅(t-ε t-s,  √(2 )θ    t+ε t )})_σ^2, β( M̅_ε t+s≥  -ε t)  d s.



Then applying (<ref>), we have 

    lim_ε→ 0lim inf_t →∞1/tlnu̅(t-s,  √(2 )θ   t+ε t )≥  1-  θ^2.




uniformly for s∈[0,1]. We further  notice  that 
 
    _σ^2, β( M̅_ε t+s≥ -ε t)
      → 1,  as t→∞,

uniformly for s∈[0,1]. Then applying Jensen's inequality and Fatou's lemma, one has

    lim inf_t →∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2 )θ      t) 
         ≥ 
      1- θ^2.

This proves Theorem <ref> for the cases that  β >1, or β≤ 1, σ ^2 ≤   1, or β≤ 1, σ ^2 > 1 and θ^2<(β -1) /(1-σ^2), since  1- θ^2≥β-θ^2/σ^2 in all above cases.




  §.§.§ The case of max_0≤ u≤ 1 f_2(u)=f_2(1). 
 

This case corresponds to the case that
β≤ 1, 1<σ ^2 ≤1/β and θ^2≥(1-β) /(σ^2-1) σ ^4. Again, we only give an outline of the proof. Choose 

    I_t=[(1-ε) t, (1-ε) t+1],   x_t=√(2 )θ t+ε t,

for  ε>0 small enough.  Then we have

    (∃  u ∈_t^2: X_t^u ⩾√(2 )θ  t) 
        ≥∫_0^1  ( 1- e^-αu̅(ε t-s, -ε t )|_(1-ε) t+s ^1(x_t)  | ≥ 1 )_σ^2, β( M̅_(1-ε) t+s > √(2 )θ t+ε t )  d s.


Since 

    lim_ε→0lim inf_t→∞1/tln_σ^2, β( M_(1-ε) t+s > √(2 )θ t+ε t  )
     ≥β -θ ^2/σ ^2,

then we have

    lim inf_t→∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2)θ  t)≥β -θ ^2/σ ^2.

 This proves Theorem <ref> for the case that
β≤ 1, 1<σ ^2 ≤1/β and θ^2≥(1-β) /(σ^2-1) σ ^4.



  §.§.§ The case of max_0≤ u≤ 1 f_2(u)=f_2(u_0) for 0<u_0<1. 


Notice that  f_2'(u_0)=0.  This  corresponds to  the case that
β <1,  1<σ ^2 < 1/β, and 

    (1-β) /(σ^2-1)  <   θ^2 <  (1-β) /(σ^2-1) σ ^4.

We also recall that in this case

    0<u_0=√(σ^2-1/1-β)·θ  -1/σ^2-1<1

and

    f_2(u_0)=-2 √((1-β)(σ^2-1))θ-β+σ^2/σ^2-1.




Choose 

    x_0=√(2 )θ u_0 σ^2/1-u_0+u_0 σ ^2,   x_t=x_0 t,  β_0=  β u_0 -x_0 ^2/2 σ^2 u_0>0.


One can check that x_0<√(2)θ since u_0>0.
Then,

    (∃  u ∈_t^2: X_t^u ⩾√(2)θ   t) 
        ≥∫_0^1  (1-e^-α   e^(β_0-ε) tu̅(t-u_0t-s, √(2)θ  t-x_0 t ))  ( |_u_0t+s^1(x_t)  | ≥ e^(β_0-ε) t)  d s.






We also notice that

    (√(2)θ-x_0)^2/2(1-u_0)^2>1,

since

    σ^2<1/β<2-β,  for  β<1.

 
 Then applying (<ref>) again, we have 

    lim inf_t →∞1/tlnu̅(t-u_0 t-s, √(2 )θ t-x_0 t )
     ≥(1-(√(2)θ-x_0)^2/2(1-u_0)^2)(1-u_0),

uniform for s∈ [0,1].  Furthermore, according to (<ref>), 

    1/tlog |_u_0 t^1(x_t) |→β_0,   a.s.

Thus

    lim_t→∞( |_u_0 t+s^1(x_t)  | ≥ e^(β_0-ε) t) = 1,

uniform for s∈ (0,1). Then one has

    lim inf_t →∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2)θ   t) 
        ≥β_0+(1-(√(2)θ-x_0)^2/2(1-u_0)^2)(1-u_0)
         = (β-1) u_0+1-θ^2/(σ^2-1) u_0+1     = 
    -2 √((1-β)(σ^2-1))θ-β+σ^2/σ^2-1   ≥β-θ^2/σ^2=f_2(1),

where the last inequality holds for the case that β <1, σ ^2> 1, and 

    (1-β) /(σ^2-1) ≤θ^2 ≤(1-β) /(σ^2-1) σ ^4.


This proves Theorem <ref> for the above case. 


  






§ PROOF OF THEOREM <REF>


In this section, we always assume that

    (β,σ^2)∈𝒞_3  ={(β, σ^2): σ^2+β>2  and σ^2<β/2 β-1}.

Recall ξ =σ ^2-β/√(2(1-σ ^2)(β -1)).
The arguments are also exactly the same to those in Section <ref>. One just to replace θ by ξθ/√(2). Again, We only give an outline here. 



 §.§ Upper bound


First, by union bound, one has

    (M_t ⩾ξθ t) ⩽   2max( e^β t(σ B_t ⩾ξθ t),α∫_0^t e^β s+(t-s)(σ B_s+B_t-B_s ⩾ξθ t)  d s)
    =:     2max(
    1, 2).


According to (<ref>), we have

    1≤ O(1) ·1/√(t)· e^β t· e^-ξ ^2θ^2 t^2/2 σ ^2t =O(1)/√(t)· e^(β-ξ ^2θ^2 /2 σ^2) t

and

    2≤α∫_0^t e^β s+(t-s)·1/√(2 π(σ^2 s+t-s))· e^-ξ ^2 θ^2 t^2/2(σ^2 s+t-s).



For 2, set

    f_3(u)=(β-1) u+1-ξ^2 θ^2/2((.σ^2-1) u+1),    u ∈(0,1).

Then

    f_3^'(u)=β-1+ξ^2 (σ^2-1)θ^2/2[(σ^2-1) u+1]^2.

Solving f^'(u_0)=0 yields


    u_0=ξθ√(1-σ^2/2(β-1)) -1/σ^2-1


and

    f_3(u_0)=√(2)(1-β)-θ^2(σ^2-β)^2-1/√(2)(σ^2-1)+1.



Since (β, σ^2) ∈𝒞_3, then β >1,σ ^2 < 1 and hence u↦ f_3^'(u) is decreasing. We have three cases. 

If θ^2≥2(β-1)/1-σ^21/ξ^2, then f_3^'(0)≤  0. Thus max_0≤ u≤ 1f_3(u)=f_3(0) and hence

    lim sup_t →∞1/tln(M_t>ξθ t)≤ f_3(0)=  1-ξ^2θ^2/2.


If θ^2≤2(β-1)/1-σ^2σ^4/ξ^2, then f^'(1)≥ 0. Thus max_0≤ u≤ 1f_3(u)=f_3(1) and hence

    lim sup_t →∞1/tln(M_t>ξθ t)≤ f_3(1)=   β-ξ^2 θ^2/2 σ^2.


If 2(β-1)/1-σ^2σ^4/ξ^2 < θ^2< 2(β-1)/1-σ^21/ξ^2, then f_3^'(0)>0  and f_3^'(1)<0. Thus max_0≤ u≤ 1f_3(u)=f_3(u_0) and hence

    lim sup_t →∞1/tln P(M_t>ξθ t)≤ f_3(u_0)=  √(2)(1-β)-θ^2(σ^2-β)^2-1/√(2)(σ^2-1)+1.







 §.§ Lower bound for Theorem <ref>


Notice that

    (M_t ⩾ξθ  t) 
    ⩾max{(∃  u ∈_t^1: X_t^u ⩾ξθ  t), (∃  u ∈_t^2: X_t^u ⩾ξθ  t)}.


First, from <ref>, we have

    lim_t→∞1/tln(∃  u ∈_t^1: X_t^u ⩾ξθ   t ) 
        =lim_t→∞1/tln_σ^2, β(M̅_t ⩾ξθ  t)     = β-ξ^2θ^2/2σ^2.


By similar reasoning as (<ref>), (<ref>) and (<ref>), we have

    (∃  u ∈_t^2: X_t^u ⩾ξθ   t)
           ≥∫_I_t1-   [exp{-α∑_u∈_s^1(x_t)u̅(t-s, ξθ  t-x_t )}]  d s,

where I_t⊂ [0, t] is an interval, x_t∈ℝ is a deterministic function of t. 
Both I_t and x_t will be determined later from case to case. Recall from (<ref>) that

    f_3(u)=(β-1) u+1-ξ^2 θ^2/2((.σ^2-1) u+1),    u ∈(0,1).





  §.§.§ The case of max_0≤ u≤ 1 f_3(u)=f_3(0). 



This corresponds to the cases that 
θ^2≥2(β-1)/1-σ^21/ξ^2.  Choose 

    I_t=[ε t, ε t+1],   x_t=-ε t,

for  ε>0 small enough. Then 

    (∃  u ∈_t^2: X_t^u ⩾ξθ   t)    ≥∫_0^1  (1- exp{-αu̅(t-ε t-s,  ξθ    t+ε t )})_σ^2, β( M̅_ε t+s≥  -ε t)  d s.



Then applying (<ref>), we have 

    lim_ε→ 0lim inf_t →∞1/tlnu̅(t-s,  ξθ   t+ε t )≥  1-  ξ^2θ^2/2.




We further  notice  that 
 
    _σ^2, β( M̅_ε t+s≥ -ε t)
      → 1,  as t→∞,

Then applying Jensen's inequality and Fatou's lemma, one has

    lim inf_t →∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2 )θ      t) 
         ≥ 
     1-  ξ^2θ^2/2.

This proves Theorem <ref> for the cases that  θ^2≥2(β-1)/1-σ^21/ξ^2, since  1-  ξ^2θ^2/2≥β-ξ^2θ^2/2 σ^2 in this case.




  §.§.§ The case of max_0≤ u≤ 1 f_2(u)=f_2(1). 
 

This case corresponds to the case that
 θ^2≤2(β-1)/1-σ^2σ^4/ξ^2. Choose 

    I_t=[(1-ε) t, (1-ε) t+1],   x_t=√(2 )θ t+ε t,

for  ε>0 small enough.  Then we have

    (∃  u ∈_t^2: X_t^u ⩾ξθ  t) 
        ≥∫_0^1  ( 1- e^-αu̅(ε t-s, -ε t )|_(1-ε) t+s ^1(x_t)  | ≥ 1 )_σ^2, β( M̅_(1-ε) t+s > ξθ t+ε t )  d s.


Since 

    lim_ε→0lim inf_t→∞1/tln_σ^2, β( M̅_(1-ε) t+s > ξθ t+ε t  )
     = β -ξ^2θ^2/2σ ^2,

then we have

    lim inf_t→∞1/tln(∃  u ∈_t^2: X_t^u ⩾√(2)θ  t)= β -ξ^2θ^2/2σ ^2.

 This proves Theorem <ref> for the case that θ^2≤2(β-1)/1-σ^2σ^4/ξ^2.



  §.§.§ The case of max_0≤ u≤ 1 f_3(u)=f_3(u_0) for 0<u_0<1. 


Notice that  f_3'(u_0)=0.  This  corresponds to  the case that
 2(β-1)/1-σ^2σ^4/ξ^2 <θ^2<  2(β-1)/1-σ^21/ξ^2. Choose 

    x_0=ξθ u_0 σ^2/1-u_0+u_0 σ ^2,   x_t=x_0 t,  β_0=  β u_0 -x_0 ^2/2 σ^2 u_0>0.


Obviously x_0<ξθ since u_0>0.
Then

    (∃  u ∈_t^2: X_t^u ⩾ξθ   t) 
        ≥∫_0^1  (1-e^-α   e^(β_0-ε) tu̅(t-u_0t-s, ξθ  t-x_0 t ))  ( |_u_0t+s^1(x_t)  | ≥ e^(β_0-ε) t)  d s.






We also notice that

    (ξθ-x_0)^2/2(1-u_0)^2>1.


 
 Then applying (<ref>) again, we have 

    lim inf_t →∞1/tlnu̅(t-u_0 t-s, ξθ t-x_0 t )
     ≥(1-(ξθ-x_0)^2/2(1-u_0)^2)(1-u_0),

uniform for s∈ [0,1].  Furthermore, according to (<ref>), 

    1/tlog |_u_0 t^1(x_t) |→β_0,   a.s.

Thus

    lim_t→∞( |_u_0 t+s^1(x_t)  | ≥ e^(β_0-ε) t) = 1,

uniform for s∈ (0,1). Then one has

    lim inf_t →∞1/tln(∃  u ∈_t^2: X_t^u ⩾ξθ   t) 
        = 
    f_3(u_0)
       ≥β-ξ^2θ^2/2σ^2 =f_3(1),

where the last inequality holds for the case that 2(β-1)/1-σ^2σ^4/ξ^2 <θ^2<  2(β-1)/1-σ^21/ξ^2.
 This proves Theorem <ref> for the above case. We are done.


  




abbrv


Hui He

School of Mathematical Sciences, Beijing Normal University,
Beijing 100875, People's Republic of China.

E-mail: hehui@bnu.edu.cn





