Density fluctuations for the multi-species stirring process

TL;DR

This paper analyzes the equilibrium density fluctuations of a multi-species stirring process, revealing that species interactions emerge at the fluctuation level despite independent diffusion at the hydrodynamic scale.

Contribution

It introduces a multi-species generalization of the stirring process and characterizes the equilibrium fluctuations as coupled Ornstein-Uhlenbeck processes, including a reaction term extension.

Findings

Fluctuations form a coupled infinite-dimensional Ornstein-Uhlenbeck process.

Species interactions appear at the fluctuation level, not in the hydrodynamic limit.

The techniques combine Dynkin martingale approach with duality for covariance computation.

Abstract

We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein-Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.