Propagation of chaos for symmetric simple exclusions

Abstract

Notation and SummaryThe symmetric simple exclusion process (SSEP) is a continuous time Markov process in which particles perform symmetric random walks on a lattice but are excluded to occupy the same site. It is one of the simplest lattice gas models that exhibits some of the behavior expected for a large class of interacting particle systems. The object of this article is to establish propagation of chaos type behavior for SSEP. Roughly speaking, we show that if particles are initially located independently on a lattice, then, despite interaction among them, the independence is restored asymptotically at later times. We assume that initially particles are distributed identically and we show that the evolution of a single marked particle is governed by an inhomogeneous Markov process that can be uniquely determined in terms of the initial distribution and certain transport coeflcients.To define SSEP we have to determine its state space and infinitesimal generator. Let ZL denote the periodic lattice (0, 1,. . . , t} with zero and L identified.We define Zd, as the product of d copies of ZL. Let EL,N denote the space of configurations x = (XI,. . . , x~) with xi E Zf. We think of xi as the location of the i-th particle and we sometimes refer to the index of xi as the label of the particle.Suppose ( p ( z ) : z E Zd) is a symmetric probability density function of finite range; that is to say, c.. For each ( N , L ) , let ,LLN,L be a probability measure on EN,L. Throughout this paper we assume:

Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes

“…Also, we note, in [17], a "propagation of chaos" type nonequilibrium result was shown for finite-range symmetric d ≥ 1 dimensional simple exclusion processes on Z d which gives the fluctuations for a tagged particle selected at random, or in other words the average tagged particle position; however, this result, which makes key use of the "averaging," does not convey the fluctuations of any fixed, given particle and so is weaker than the one we state in this paper. In addition, with respect to certain interacting Brownian motions in one dimension, the nonequilibrium behavior was found in [7].…”

Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps

“…We already know from [11] that the scaling limit of a tagged particle is not in general a martingale. Therefore, it is not clear if this strategy can be followed for the simple exclusion process.…”

Large‐Deviation Principle for Interacting Brownian Motions

“…The hydrodynamic limit theory for f Q N g 1 N D1 is implied by the results of [10]. It is well-known [26,31] that if the scaling limit of the tagged particle is the diffusion with the generator L, which may depend on the limiting particle density , and any two tagged particles are asymptotically independent, then the limiting particle density c of each color is the unique weak solution of the PDE @ t c D L c with the initial condition 0 c .dx/ for each color c. The scaling limit as well as the asymptotic independence of the tagged particle of our system has been studied [10]. The scaling limit turned out to be the diffusion with the time-dependent generator…”

“…/ , which can be regarded as the averaged tagged particle. In the 1990s, Quastel, Rezakhanlou, and Varadhan found a systematic approach to study the limit theory and large-deviation theory for the empirical process, which resulted in a series of published work [23,24,26,34]. Despite the robustness of their methodology, the LDP for this context is only known for two models: the SSEP in the case where d 2 and the ZRP.…”