Reversible Markov processes on general spaces and spatial migration processes

Abstract

In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes.