Abstract
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.
Highlights
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
Summary
The analysis of the speed of the convergence to equilibrium for Markov processes is a major topic in probability theory. 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 [17, § 4.1] Another simple model for which the quantitative ergodicity could be investigated is the inclusion process (SIP), this model is self-dual and a coupling with independent random walks has been constructed in [21]. 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 μPt with respect to πρ. In view of [20], such decay would imply a quantitative decay on the ddistance between μPt and πρ
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