From the text: This note contains a survey of recent results concerning asymptotic properties of Markov operators and semigroups. Some biological and physical applications are given.
The organization of the paper is as follows. Section 2 contains the definitions of a Markov operator and a Markov semigroup and some examples of them. In the next section we study asymptotic properties of Markov operators and semigorups: asymptotic stability and sweeping. Theorems concerning 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 Khasminskij function. This notion is very useful in proofs of asymptotic stability of Markov semigorups. In Section 4 we give some applications of the general results to differential equations connected with diffusion and jump processes.