Mathematics – Probability
Scientific paper
2012-02-06
Mathematics
Probability
arXiv admin note: text overlap with arXiv:0811.2180
Scientific paper
We are interested by a one dimensional Markov process which moves following a diffusion for some random time and then jumps. It can represent some natural phenomena like size of cell or data transmission over the Internet. The paper begin with some results about Lipschitz contraction of semigroup. Our approach is connected with the notion of curvature introduced by Ollivier and Joulin. Our main results for jump-diffusions are quantitative estimates in Wasserstein distance, when the jump times depend of the space motion. We use different techniques which are a particular Feynmann-Kac interpretation and a non coalescent coupling. Several examples and applications are developed, including explicit formulas for the equilibrium, application to branching measure-valued processes and integrals of compound Poisson process with respect to a Brownian motion.
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