Quantitative ergodicity for the symmetric exclusion process with stationary initial data

We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.


Introduction
The analysis of the speed of the convergence to equilibrium for Markov processes is a major topic in probability theory. Referring to [15] for a general overview, we focus the discussion to the case of reversible stochastic lattice gases, i.e. conservative interacting particles systems satisfying the detailed balance condition with respect to a Gibbs measure. If these processes are considered on a bounded subset Λ of the d-dimensional lattice they are ergodic when restricted to the configurations with fixed number of particles and the corresponding reversible measure is the finite volume canonical Gibbs measure. In the high temperature regime, in [7,8,9,10,19] it has been shown that both the inverse of the spectral gap and the logarithmic Sobolev constant grow as the square of the diameter of Λ. On the infinite lattice, stochastic lattice gases are reversible with respect to the (infinite volume) canonical Gibbs measures, see [11,Thm. 2.14]. In the high temperature regime, by [11,Thm. 5.14], the extremal elements of the set of the canonical Gibbs measures consist in the one parameter family {π ρ } where π ρ is the grand-canonical Gibbs measure with density, i.e. expected number of particles per site, given by ρ. Moreover, as follows from [11,Thm. 1.72], the semigroup P t , t ≥ 0 associated to a reversible stochastic lattice gases is ergodic in L 2 (π ρ ) namely, P t f − π ρ (f ) L2(πρ) → 0 for each f ∈ L 2 (π ρ ). A quantitative version of this statement can be obtained when the function f is local, i.e. it depends on the particles configuration only through its value on finitely many sites. For this class of functions it has been shown, for the exclusion and the zero range processes, that P t f − π ρ (f ) 2 L2(πρ) ≤ C t −d/2 for some constant C = C(f ) [3,12]. The case in which the reversible probability is a grand-canonical Gibbs measure in the high temperature regime is discussed in [4,6,14] where a slightly worse bound is proven.
We here consider the simple symmetric exclusion process. It corresponds to the infinite temperature case and the probability measure π ρ is the product Bernoulli measure with parameter ρ ∈ [0, 1]. If the probability µ is a suitable local perturbation of π ρ it has been proven in [3] that Ent µP t π ρ ≤ Ct −d/2 for some constant C = C(µ), here Ent denotes the relative entropy. See also [18] for further details on this issue. In general, it appears to be quite difficult to characterize the probabilities µ on the configuration space such that µP t converges to π ρ as t → ∞. However, as proven in [17, Thm. VIII.1.47], such convergence holds when µ is stationary, i.e. invariant with respect to space shifts, and ergodic with density ρ. Our purpose is to provide a quantitative version of this statement. More precisely, denoting byd the Ornstein distance on the set of stationary probabilities [22, § I.9.b], we prove here that if µ is stationary, ergodic with density ρ, and has absolutely summable correlations, thend(µP t , π ρ ) ≤ Ct −γ(d) for some constant C = C(µ) and γ(d) = d/4 for d < 4 and γ(d) = 1 for d ≥ 4; see Theorem 2.1 below. The proof is achieved by combining a simple coupling argument with the analysis on the decay of the density for the two-species symmetric exclusion process with annihilation [1,2], that relies on an analogous result for the two-species independent random walks [5].
Referring to [5] for more details, we next explain heuristically the power law decay of the Ornsteind distance. By [22,Thm. I.9.7] thed distance between µP t and π ρ can be bounded using a coupling between µP t and π ρ invariant with respect to space shifts:d(µP t , π ρ ) ≤ P η 0 (t) = ζ 0 (t) . Here (η 0 (t), ζ 0 (t)) are the occupation numbers at the origin at time t ≥ 0 of a two-species annihilating exclusion process with equal density. Namely, two species of particles that evolve on Z d according to exclusion processes and annihilating when they meet. Let ρ(t) = P η 0 (t) = ζ 0 (t) be the probability that the origin is occupied by either species of particles. In the mean field approximation, ρ(t) decays to zero according toρ(t) = −ρ(t) 2 which would implyd(µP t , π ρ ) ≈ t −1 . This approximation yields the correct behavior when d ≥ 4 while for d ≤ 3 the Gaussian fluctuations of the initial data become relevant and, due to the underlying particle's diffusion, the decay becomesd(µP t , π ρ ) ≈ t −d/4 .
To our knowledge, the present analysis of the symmetric exclusion process is the first example in which the quantitative ergodicity for a stochastic lattice gas with stationary initial data has been achieved. The arguments here developed cover directly the case of independent random walks. We conclude by discussing possible extensions to other models. As mentioned before, the crucial ingredient in the proof is the quantitative decay of the density for the two-species exclusion process with annihilation. This decay might be proven for other attractive stochastic lattice gases such as the zero range process with increasing rates, see e.g. [13,Thm. 2.5.2], or the special class of reversible stochastic lattice gases in [16, § 4.1]. Another simple model for which the quantitative ergodicity could be investigated is the inclusion process (SIP), indeed this model is self-dual and a coupling with independent random walks has been constructed in [21]. For the generic case of reversible stochastic lattice gases where the invariant measure is not a product measure, it seems however difficult that coupling arguments suffice to establish the quantitative ergodicity, cfr. the corresponding problem of the decay to equilibrium for local functions in [4,6,12,14]. We remark that another possible setting to discuss the quantitative ergodicity for stochastic lattice gases with stationary initial data µ is the decay rate of the relative entropy per site of µP t with respect to π ρ . In view of [20], such decay would imply a quantitative decay on thed distance between µP t and π ρ .

Notation and results
Let Z d be the d-dimensional lattice. We write Λ ⊂⊂ Z d when Λ is a finite subset of Z d . Set Ω := {0, 1} Z d that it is considered endowed with the product topology and the corresponding Borel σ-algebra. Elements of Z d will be called sites while elements of Ω configurations. For η ∈ Ω the value of the configuration η at the site x, denoted by η x ∈ {0, 1}, is interpreted as the absence/presence of a particle at x and called occupation number. In particular, η 0 is the occupation number at the origin of Z d .
The simple symmetric exclusion process (SEP) is the Markov process on the state space Ω whose generator acts on local functions f : Ω → R, i.e. functions depending on η only through the values {η x } for finitely many sites x, as The sum is carried out over the unordered edges of Z d and η x,y is the configuration obtained from η by exchanging the occupation numbers at x and y, We denote by P t , t ≥ 0, the semigroup generated by L that acts on the Banach space C(Ω), the family of continuous function on Ω endowed with the uniform norm. We refer to [17, Ch. VIII] for the construction of this process and its properties. In Theorem 1.44 there, it is proven in particular that a probability µ on Ω is invariant for SEP if and only if µ is exchangeable, equivalently µ is a mixture, i.e. a possibly infinite convex combination, of i.i.d. Bernoulli measures. Let P τ (Ω) be the set of stationary probabilities on Ω, i.e. the probabilities on Ω invariant with respect to the space shifts on Ω. Observe that P τ (Ω) is a convex set and the set of its extremal points, denoted by P τ,e (Ω), consists of the ergodic probabilities. For ρ ∈ [0, 1] let π ρ ∈ P τ (Ω) the Bernoulli product probability with parameter ρ. In [17, Thm. VIII.1.47 ] it is proven that if µ ∈ P τ,e (Ω) and µ(η 0 ) = ρ then µP t weakly converges to π ρ as t → +∞. The purpose of the present analysis is to provide a quantitative version of this statement with an explicit control on the rate of convergence. This will be achieved when the probability µ has absolutely summable correlations.
To formulate the quantitative ergodicity we need a distance on P τ (Ω). We shall use the so-called Ornsteind distance. Given Λ ⊂⊂ Z d let d Λ be the distance on Denoting by P(Ω Λ ) the set of probabilities on Ω Λ , let W Λ be the 1-Wasserstein distance on P(Ω Λ ) associated to d Λ , i.e.
where the infimum is carried out over all the couplings Q of µ and ν, i.e. the set of probabilities on Ω×Ω with marginals µ and ν. For µ ∈ P τ (Ω) and Λ ⊂ Z d let µ Λ be the marginal of µ on Ω Λ . By a standard super-additive argument, if µ, ν ∈ P τ (Ω) then lim Moreover, see e.g. [22,Thm. I.9.7],d defines a distance on P τ (Ω) that can be represented asd where we recall that η 0 , ζ 0 are the occupation numbers at the origin and the infimum is carried out over all the stationary couplings Q of µ and ν, i.e. the set of couplings of µ and ν that are invariant with respect to space shifts on Ω × Ω. By (2.3), the topology induced byd on P τ (Ω) is finer than the topology induced by the weak convergence. Denoting by ent(µ|ν) the relative entropy per unit of volume of µ with respect to ν, we finally mention two remarkable properties of the Ornsteind distance, see e.g. [22, Thm. I.9.15 and I.9.16]: (i) P τ,e (Ω) isd closed, (ii) for each ρ ∈ [0, 1] the map P τ,e (Ω) ∋ µ → ent(µ|π ρ ) isd continuous.
Given two functions f, g and a probability µ we let µ(f ; g) := µ(f g) − µ(f )µ(g) be the µ-covariance of f and g. For µ ∈ P τ,e (Ω) we set (2.5) The quantitative ergodicity for SEP with stationary initial data is then stated as follows.

Reduction to the two species SEP with annihilation
In view of (2.4), an upper bound ford(µP t , π ρ ) can be obtained by exhibiting a stationary coupling between µP t and π ρ . Starting a time t = 0 by a stationary coupling of µ and π ρ and coupling the corresponding two SEP we obtain, at time t > 0, a stationary coupling between µP t and π ρ good enough to produce the bound (2.6). In words, the coupling between the two SEP can be described as follows. Particles of the two processes that are at the same site jump together while particles alone jump independently. In formulae, we consider the Markov process whose generator is the closure of the operator L that acts on local functions F : Ω × Ω → R as The corresponding semigroup is denoted by P t , t ≥ 0. Note that, even if not apparent from the notation, L is the operator used in [17, § VIII.2] to prove the attractiveness of the exclusion process. The next statement can be checked by a direct computation that is omitted.
where ξ x,y has been defined in (2.2) and The process generated by (3.2), that will be referred to as the two species SEP with annihilation, can be described by visualizing the sites x where ξ x = −1 as occupied by anti-particles and the sites x where ξ x = 1 occupied by particles. Particles and anti-particles evolve following two independent SEP and when a particle jumps over a anti-particle, or conversely a anti-particle jumps over a particle, the two particles are annihilated. It can be therefore seen as kinetic model corresponding to the reaction anti-particle + particle → ∅. As we show in the next section, an analysis of this dynamics yields an upper bound for the Ornstein distance between two SEP with different initial data.

Long time behavior of the two species SEP with annihilation
In this section we consider the two species SEP with annihilation obtaining -for suitable stationary initial data -an upper bound for the probability that at time t > 0 the origin is occupied by either types of particles. Given a probability ℘ on S the law of the two species SEP with annihilation, i.e. the process generated by (3.2), and initial datum ℘ is denoted by P ℘ , the corresponding expectation by E ℘ . For ℘ ∈ P τ,e (S), the set of stationary and ergodic probabilities on S, we set Theorem 4.1. For each d there exists a constant C such that for any t > 0 and any ℘ ∈ P τ,e (S) satisfying ℘(ξ 0 = −1) = ℘(ξ 0 = 1) The analogous statement for two species annihilating independent random walks and stationary product initial condition has been proven in [5]. Relying on the arguments there, the bound (4.2) is proven in [1,2] when the initial datum ℘ is a product measure. This assumption on the initial datum is used only in [1, Lemma 2.1]; however, as we show in Lemma 4.2 below, it can be relaxed to the condition B(℘) < +∞. The rest of the arguments in [1,2] carries out unchanged to the present setting and yields the statement in Theorem 4.1. Assuming it, we first conclude the proof of the quantitative ergodicity for the SEP with stationary initial data.
In order to extend the result in [1,2] to non-product initial data, we need to realize the two species SEP with annihilation on the probability space associated to the so-called stirring process. This construction is achieved in two steps: from two independent stirring processes we first obtain the two species SEP without annihilation then, by a thinning procedure, we construct the the two species SEP with annihilation.
We start by recalling the graphical construction of the stirring process. To each site x ∈ Z d attach a copy of the positive half-axis R + . For each edge x, y draw a set of double-arrows sampled according to independent Poisson point processes with intensity one. The stirring process W = {W x (t), x ∈ Z d , t ∈ R + } is defined as follows: the value W x (t) ∈ Z d is obtained by placing a marker at time t = 0 at the point x and letting it evolve following the path dictated by the arrows. Given ζ ∈ {0, 1} Z d the SEP with initial datum ζ can be realized as η x (t) = y∈Z d ζ y 1 {x} (W y (t)), x ∈ Z d . Let finally W = W − , W + be two independent copies of the stirring process.
The two species SEP without annihilation can be described as follows. Each site can be: empty, occupied by a particle, occupied by a anti-particle, or occupied by both a particle and an anti-particle. The anti-particles evolve according to the stirring process W − while particles according to W + . Setting S := {0, −1, +1, ±} Z d , the two species SEP without annihilation is thus the process on the state space S defined by Then ξ(t), t ≥ 0, is the Markov process whose generator L acts on local functions f : S → R as where the leftmost sum is carried out over the set of oriented edges of Z d , ξ x,y has been defined in (2.2) and, given α, β ∈ {0, −1, +1, ±}, Given a probability ℘ on S we denote by P ℘ the law of this process with initial condition ℘ and by E ℘ the corresponding expectation.
The two species SEP with annihilation ξ(t), t ≥ 0, can be finally realized by a thinning of two species SEP without annihilation by recursively removing pairs of particles of different species that occupy the same site. This thinning procedure provides a coupling of the processes ξ(t) and ξ(t) such that for any t ≥ 0 and α ∈ {−1, 1} we have {x ∈ Z d : ξ t (x) = α} ⊂ {x ∈ Z d : ξ t (x) = α} with probability one.