Densities for piecewise deterministic Markov processes with boundary

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:479

Article

Gwizdz, Piotr¹  Tyran-Kaminska, Marta^1,2 

[1] Univ Silesia, Inst Math, Bankowa 14, PL-40007 Katowice, Poland

[2] Polish Acad Sci, Inst Math, Bankowa 14, PL-40007 Katowice, Poland

关键词: Substochastic semigroup;    Invariant density;    Perturbation of boundary conditions;    Initial-boundary value problem;    Transport equation;    Cell cycle model;   

DOI  :  10.1016/j.jmaa.2019.06.032

来源: Elsevier

【 摘 要 】

We study the existence of densities for distributions of piecewise deterministic Markov processes. We also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times. In our approach we use functional-analytic methods and the theory of linear operator semigroups. By imposing general conditions on the characteristics of a given Markov process, we show the existence of a substochastic semigroup describing the evolution of densities for the process and we identify its generator. Our main tool is a new perturbation theorem for substochastic semigroups, where we perturb both the action of the generator and of its domain, allowing to treat general transport-type equations with non-local boundary conditions. A couple of particular examples illustrate our general results. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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