Exponential ergodicity for damping Hamiltonian dynamics with state-dependent and non-local collisions

TL;DR

This paper proves exponential ergodicity in Wasserstein distance for a class of damping Hamiltonian dynamics with state-dependent, non-local collisions, using advanced coupling techniques, extending previous results on stochastic Hamiltonian systems and Andersen dynamics.

Contribution

It introduces a novel approach combining refined basic and reflection couplings to establish exponential ergodicity for complex non-local Hamiltonian systems.

Findings

Proves exponential ergodicity in Wasserstein distance for the model.

Extends previous ergodicity results to systems with state-dependent, non-local collisions.

Employs advanced coupling methods to handle non-local operators.

Abstract

In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes while is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this work is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present paper is a continuation of the counterpart in \cite{BW2022} on exponential ergodicity of stochastic Hamiltonian systems with L\'evy noises and a complement of \cite{BA} upon exponential ergodicity for Andersen dynamics with constant jump rate functions.