Markov Processes

Abstract

This chapter presents the fundamentals of the theory of general Markov processes in continuous time. Section 21.1 contains the definitions and a discussion of the Markov property and transition functions, and derives the Chapman–Kolmogorov equation. Section 21.2 studies Markov processes in countable state spaces, deriving systems of backward and forward differential equations for transition probabilities. It also establishes the ergodic theorem and contains examples illustrating the presented theory. Section 21.3 deals with continuous time branching processes. Then the elements of the general theory of semi-Markov processes are presented in Sect. 21.4, including the ergodic theorem and some other related results for such processes. Section 21.5 discusses the so-called regenerative processes, establishing their ergodicity and the Laws of Large Numbers and Central Limit Theorem for integrals of functions of their trajectories. Section 21.6 is devoted to diffusion processes. It begins with the classical definition of diffusion, derives the forward and backward Kolmogorov equations for the transition probability function of a diffusion process, and gives a couple of examples of using the equations to compute important characteristics of the respective processes.