Substochastic semigroups and densities of piecewise deterministic Markov processes

Densities for piecewise deterministic Markov processes with boundary

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 sub…

Existence of invariant densities for semiflows with jumps

The problem of existence and uniqueness of absolutely continuous invariant measures for a class of piecewise deterministic Markov processes is investigated using the theory of substochastic semigroups obtained through the Kato--Voigt perturbation theorem on the $L^1$-space. We provide a new criterion for the existence of a strictly positive and unique invariant density for such processes. The long time qualitative behavior of the corresponding semigroups is also considered. To illustrate our general results we give a detailed study of a two dimensional model of gene expression with bursting.

Transport equations and perturbations of boundary conditions

We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to non-densely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.

通过提升与流庞加莱不等式研究非可逆马尔可夫过程的收敛性

Convergence of non-reversible Markov processes via lifting and flow Poincar{é} inequality

本文提出基于二阶提升和亚强制性变分方法的通用框架，引入流庞加莱不等式与一般散度引理，用于非可逆马尔可夫过程的定量收敛分析，并成功应用于非平衡统计力学模型和带一般随机跳跃核的分段确定性采样过程。