Multiparameter Markov Processes

Abstract

We can informally interpret Chapter 8’s definition of a Markov process X = (X _t ; t ≥ 0) as a (one-parameter) process whose “future” values X _t+s depend on the past only through its current value X _t . While this is perfectly intuitively clear (due to the well-ordering of the “time axis”), it is far less clear what a multiparameter Markov process should be. In this chapter we introduce and study a class of random fields called multiparameter Markov processes. The definitions, given early on, are motivated by the potential theory that is developed later in this chapter. We will also see how this multiparameter theory can be used to study intersections of ordinary one-parameter processes.