Markov operators: applications to diffusion processes and population dynamics

Abstract

To the memory of Wies law Szlenk Abstract. This note contains a survey of recent results concerning asymp- totic properties of Markov operators and semigroups. Some biological and physical applications are given. 1. Introduction. Dynamical systems and dynamical systems with stochastic perturbations can be effectively studied using Markov operators and Markov semigroups. Semigroups of Markov operators are generated by partial differential equations (transport equations). Equations of this type appear in the theory of diffusion processes and in population dynamics. In this note we present new results in the theory of Markov operators and illustrate them by some biological and physical applications. The results presented are based on the papers (16-18, 22). The organization of the paper is as follows. Section 2 contains the defi- nitions of a Markov operator and a Markov semigroup and some examples of them. In the next section we study asymptotic properties of Markov op- erators and semigroups: asymptotic stability and sweeping. Theorems con- cerning asymptotic stability and sweeping allow us to formulate the Foguel alternative. This alternative says that under suitable conditions a Markov operator (semigroup) is asymptotically stable or sweeping. Then we define a new notion called a Hasminskiuo function. This notion is very useful in proofs of asymptotic stability of Markov semigroups. In Section 4 we give some applications of the general results to differential equations connected with diffusion and jump processes.