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

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 determinis-tic Markov processes that 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 continua-tion of the counterpart in (Stochastic Process. Appl. (2022) 146 114-142) on exponential ergodicity of stochastic Hamiltonian systems with Levy noises and a complement of (Ann. Inst. Henri Poincare Probab. Stat. 58 (2022a) 916-944) upon exponential ergodicity for Andersen dynamics with constant jump rate functions.