On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion

TL;DR

This paper proves a functional central limit theorem for the current and tagged particle position in the one-dimensional symmetric simple exclusion process, confirming a longstanding conjecture and strengthening previous convergence results to a stronger topology.

Contribution

It improves existing convergence results from finite-dimensional distributions to a functional CLT in the uniform topology for the simple exclusion process.

Findings

Convergence to fractional Brownian motion with Hurst parameter 1/4

Strengthened from finite-dimensional to functional convergence

Completes a conjecture in the literature on simple exclusion processes

Abstract

A well-known result with respect to the one dimensional nearest-neighbor symmetric simple exclusion process is the convergence to fractional Brownian motion with Hurst parameter 1/4, in the sense of finite-dimensional distributions, of the subdiffusively rescaled current across the origin, and the subdiffusively rescaled tagged particle position. The purpose of this note is to improve this convergence to a functional central limit theorem, with respect to the uniform topology, and so complete the solution to a conjecture in the literature with respect to simple exclusion processes.