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

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.