Applications of the Kantorovich–Rubinstein maximum principle in the theory of Markov semigroups
Volume 448 / 2007
Dissertationes Mathematicae 448 (2007), 1-59
MSC: Primary 37A30, 47D03, 60J25; Secondary 28A33, 28A80, 60J75, 45K05.
DOI: 10.4064/dm448-0-1
Abstract
We present new sufficient conditions for the asymptotic stability of Markov operators acting on the space of signed measures. Our results are based on two principles. The first one is the LaSalle invariance principle used in the theory of dynamical systems. The second is related to the Kantorovich–Rubinstein theorems concerning the properties of probability metrics. These criteria are applied to stochastically perturbed dynamical systems, a Poisson driven stochastic differential equation and a mathematical model of the cell cycle. Moreover, we discuss the problem of the asymptotic stability of solutions of a generalized version of the Tjon–Wu equation.
