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

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