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Asymptotic Properties of Stochastic Semigroups—General Results

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Piecewise Deterministic Processes in Biological Models

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSMATHMETH))

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Abstract

In the theory of stochastic processes a special role is played by results concerning the existence of invariant densities and the long-time behaviour of their distributions. These results can be formulated and proved in terms of stochastic semigroups induced by these processes. We consider two properties: asymptotic stability and sweeping. A stochastic semigroup induced by a stochastic process is asymptotically stable if the densities of one-dimensional distributions of this process converge to a unique invariant density. Sweeping is an opposite property to asymptotic stability and it means that the probability that trajectories of the process are in a set Z goes to zero. The main result presented here shows that under some conditions a substochastic semigroup can be decomposed into asymptotically stable parts and a sweeping part. This result and some irreducibility conditions allow us to formulate theorems concerning asymptotic stability, sweeping and the Foguel alternative. This alternative says that under suitable conditions a stochastic semigroup is either asymptotically stable or sweeping.

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Correspondence to Ryszard Rudnicki .

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Rudnicki, R., Tyran-Kamińska, M. (2017). Asymptotic Properties of Stochastic Semigroups—General Results. In: Piecewise Deterministic Processes in Biological Models. SpringerBriefs in Applied Sciences and Technology(). Springer, Cham. https://doi.org/10.1007/978-3-319-61295-9_5

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