Markov Processes and Feller Semigroups

This chapter is devoted to the functional analytic approach to the study of Markov processes. In Section 21.1, we summarize the basic definitions and results about Markov processes, and formulate Markov processes in terms of transition functions. From the viewpoint of functional analysis, the transition function is something more convenient than the Markov process itself (Theorem 21.1.1). Indeed, we can associate with each transition function in a natural way a family of bounded linear operators acting on the space of continuous functions on the state space, and the so-called Markov property implies that this family forms a semigroup.