Fluctuation limit for interacting diffusions with partial annihilations through membranes

TL;DR

This paper establishes a functional central limit theorem for fluctuations in a non-equilibrium two-species interacting particle system with partial annihilations at membranes, represented as a Gaussian process solving a stochastic PDE.

Contribution

It introduces a new fluctuation limit involving a stochastic PDE with a nonlinear interface term, extending previous hydrodynamic results to non-equilibrium settings.

Findings

The fluctuation process is a distribution-valued Gaussian Markov process.

The covariance includes spatial and boundary interaction components.

The Boltzmann-Gibbs principle is validated for the system.

Abstract

We study fluctuations of the empirical processes of a non-equilibrium interacting particle system consisting of two species over a domain that is recently introduced in [8] and establish its functional central limit theorem. This fluctuation limit is a distribution-valued Gaussian Markov process which can be represented as a mild solution of a stochastic partial differential equation. The drift of our fluctuation limit involves a new partial differential equation with nonlinear coupled term on the interface that characterized the hydrodynamic limit of the system. The covariance structure of the Gaussian part consists two parts, one involving the spatial motion of the particles inside the domain and other involving a boundary integral term that captures the boundary interactions between two species. The key is to show that the Boltzmann-Gibbs principle holds for our non-equilibrium system. Our proof relies on generalizing the usual correlation functions to the join correlations at two different times.