# Weak-coupling limit for ergodic environments

## TL;DR

This paper establishes an averaging principle for interacting particle systems in ergodic environments using forward Kolmogorov equations, advancing the analysis of Markov processes in continuum systems.

## Contribution

It introduces a novel approach based on forward Kolmogorov equations to prove averaging principles in continuum particle systems with ergodic environments.

## Key findings

- Ergodicity of the environment process implies averaging for coupled Fokker-Planck equations.
- The approach applies to a broad class of Markov systems in continuum.
- Provides a new method for analyzing Markov evolutions in interacting particle systems.

## Abstract

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [Kur73]. Our approach is based on the study of forward Kolmogorov equations (a.k.a. Fokker- Planck equations). We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another (equilibrium) process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.