On the Existence and Uniqueness of Stationary Distributions for Some Class of Piecewise Deterministic Markov Processes With State-Dependent Jump Intensity

Dawid Czapla

Published 2023-03-21Version 1

In this paper, we are concerned with a class of piecewise deterministic Markov processes that involve a deterministic motion punctuated by random jumps, occurring in a Poisson-like fashion with some state-dependent rate, between which, at any time, the trajectory is driven by one of the given semiflows, randomly drawn right after the jump. We prove that there is a one-to-one correspondence between stationary distributions of such processes and those of the discrete-time Markov chains given by their post-jump locations. Further, we apply this result to provide a criterion guaranteeing the existence and uniqueness of the stationary distribution in a particular case, where the post-jump locations result from the action of an iterated function system with place-dependent probabilities.