Convergence of dynamical stationary fluctuations

TL;DR

This paper introduces a general theorem for the convergence of stationary Markov processes, simplifying proofs of fluctuation limits and enabling new results in stochastic PDEs without relying on traditional principles.

Contribution

A novel black box theorem that guarantees convergence of stationary Markov processes under broad conditions, bypassing the Boltzmann-Gibbs principle and enabling analysis of complex stochastic systems.

Findings

Stationary fluctuations of the zero-range process converge to the stochastic heat equation.

Established convergence of interface models to reflected stochastic PDEs.

Provided a unified framework for analyzing convergence in stochastic processes.

Abstract

We present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. This theorem relies on a control of the resolvents of the sequence of Markov processes, and on a suitable characterization of the resolvents of the limit. One major advantage of this approach is that it circumvents the use of the Boltzmann-Gibbs principle: for instance, we deduce in a rather simple way that the stationary fluctuations of the one-dimensional zero-range process converge to the stochastic heat equation. More importantly, it allows to establish results that were probably out of reach of existing methods: using the black box result, we are able to prove that the stationary fluctuations of a discrete model of ordered interfaces, that was considered previously in the statistical physics literature, converge to a system of reflected stochastic PDEs.