Occupation Times of Exclusion Processes

Abstract

In this paper we consider exclusion processes {η _t : t ≥ 0} evolving on the one-dimensional lattice $\mathbb{Z}$, under the diffusive time scale tn ² and starting from the invariant state ν _ρ —the Bernoulli product measure of parameter ρ ∈ [0, 1]. Our goal consists in establishing the scaling limits of the additive functional $\varGamma _{t}:=\int _{ 0}^{tn^{2} }\eta _{s}(0)\, ds$—the occupation time of the origin. We present a method, recently introduced in Gonçalves and Jara (Universality of KPZ equation, Available online at arXiv:1003.4478, 2011), from which a local Boltzmann-Gibbs Principle can be derived for a general class of exclusion processes. In this case, this principle says that Γ _t is very well approximated to the additive functional of the density of particles. As a consequence, the scaling limits of Γ _t follow from the scaling limits of the density of particles. As examples we present the mean-zero exclusion, the symmetric simple exclusion and the weakly asymmetric simple exclusion. For the latter under a strong asymmetry regime, the limit of Γ _t is given in terms of the solution of the KPZ equation.