Exponential ergodicity for a class of non-Markovian stochastic processes

TL;DR

This paper establishes exponential convergence to equilibrium for certain non-Markovian stochastic differential equations with delay, using cluster expansion techniques, and extends results to small perturbations of ergodic diffusions.

Contribution

It introduces a novel application of cluster expansion methods to prove exponential ergodicity in non-Markovian stochastic processes with delay.

Findings

Proves exponential convergence to invariant measure.

Applies results to small perturbations of ergodic diffusions.

Extends ergodicity results to non-Markovian settings.

Abstract

We prove the convergence at an exponential rate towards the invariant probability measure for a class of solutions of stochastic differential equations with finite delay. This is done, in this non-Markovian setting, using the cluster expansion method, inspired from [4] or [14]. As a consequence, the results hold for small perturbations of ergodic diffusions.