Non-equilibrium fluctuations for the SSEP with a slow bond

TL;DR

This paper establishes non-equilibrium fluctuation results for the 1D symmetric simple exclusion process with a slow bond, extending previous equilibrium results through novel correlation estimates and PDE analysis.

Contribution

It introduces a new method for estimating correlations in non-homogeneous exclusion processes using PDEs and local times, applicable to various models.

Findings

Proved non-equilibrium fluctuations for SSEP with a slow bond.

Developed a PDE-based approach to correlation estimates.

Connected correlation analysis to local times of random walks.

Abstract

We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of T. Franco, A. Neumann and P. Gon\c{c}alves (2013), which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel's principle. Projection techniques and coupling arguments reduce the analysis to the problem of studying the local times of the classical random walk. We think that the method developed here can be applied to a variety of models, and we provide a discussion on this matter.