Equilibrium fluctuations for the Symmetric Exclusion Process on a compact Riemannian manifold

TL;DR

This paper studies the equilibrium fluctuations of the Symmetric Exclusion Process on a compact Riemannian manifold, demonstrating convergence to a generalized Ornstein-Uhlenbeck process, thus extending understanding of stochastic particle systems on curved spaces.

Contribution

It establishes the convergence of equilibrium fluctuations to a generalized Ornstein-Uhlenbeck process on a Riemannian manifold, extending previous hydrodynamic limit results.

Findings

Fluctuation fields converge to a generalized Ornstein-Uhlenbeck process.

Proved tightness and martingale problem for the limiting process.

Extended fluctuation analysis to curved geometric settings.

Abstract

We consider the Symmetric Exclusion Process on a compact Riemannian manifold, as introduced in van Ginkel and Redig (2020). There it was shown that the hydrodynamic limit satisfies the heat equation. In this paper we study the equilibrium fluctuations around this hydrodynamic limit. We define the fluctuation fields as functionals acting on smooth functions on the manifold and we show that they converge in distribution in the path space to a generalized Ornstein-Uhlenbeck process. This is done by proving tightness and by showing that the limiting fluctuations satisfy the corresponding martingale problem.