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

Abstract

In this paper, we consider a subclass of piecewise deterministic Markov processes with a Polish state space that involve a deterministic motion punctuated by random jumps, occurring in a Poisson-like fashion with some state-dependent rate, between which the trajectory is driven by one of the given semiflows. We prove that there is a one-to-one correspondence between stationary distributions of such processes and those of the Markov chains given by their post-jump locations. Using this result, we further establish 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 a random iterated function system with an arbitrary set of transformations.

References in corpus (4)

• Dynamic behavior of stochastic gene expression models in the presence of bursting
• Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling
• Exponential ergodicity of some Markov dynamical system with application to a Poisson driven stochastic differential equation
• Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows