Homogeneous Markov Processes

Abstract

In the previous section a general definition of a Markov process was given and the basic properties of these processes were studied. The present chapter is devoted to the most important class of Markov processes—the homogeneous (or more precisely homogeneous in time) Markov processes. Roughly speaking a Markov process is homogeneous if its transition probabilities P(t,x,s,B) depends only on the difference s — t. However, in the modern theory of Markov processes a more restrictive definition is given. This definition involves certain restrictions on the set of sample functions of the processes also. We shall, however, utilize an even more restrictive definition which is convenient for solving basic problems of the theory and which is at the same time a natural definition for the construction of a process with a given transition probability. The basic difference between the definition presented below and the general definition is that here we consider Markov processes with respect to a current of σ-algebras generated by the values of the sample functions of the process and not with respect to an arbitrary current of σ-algebras.