Identifying Markov Processes Up to Time Change

Abstract

The intertwining of Markov processes and potential theory has been apparent at least since Hunt’s fundamental trilogy on these subjects and was certainly evident even before then. The relationships between these two subjects have been investigated vigorously and profitably since then, and we intend to add to this study here. The central object of interest in potential theory is the cone of excessive functions, a positive cone of functions satisfying various axioms or principles of potential theory (see for example [7] and the references). The best known is the cone of superharmonic functions in R³ consisting of positive constants together with functions of the form ∫ |x-y|^-1 μ(dy), where µ is a positive measure: this arises in Newtonian potential theory, and today’s axiomatic approach owes a great deal to abstraction and generalization of properties of this particular cone of functions. Each reasonable Markov process (X(t), P^x) on E has an associated cone of excessive functions S(X) which can be obtained in an analytic manner from the semigroup P(t): throw a positive function f(x) into the cone if P(t)f(x) ≤ f(x) for all positive t and if P(t)f(x) increases to f(x) as t decreases to zero. Such a cone may contain only constant functions. We restrict our detailed discussion to transient processes (see (1.1) for a definition) so that the excessive functions separate points in the state space E. Later in this section, we discuss the non-transient case briefly.

Research supported in part by NSF Grant MCS-8002659.