Isomorphism Theorems: Markov Processes, Gaussian Processes and Beyond

Abstract

We begin with some introductory material on Gaussian processes and Markov processes and then study in turn the isomorphism theorems of Dynkin, Eisenbaum and the generalized second Ray-Knight theorem. In each case we give a proof and a sample application. We then introduce loop soups and permanental processes, the ingredients we use to develop an isomorphism theorem for non-symmetric Markov processes. Along the way we gain new insight into the reason that Gaussian processes appear in the isomorphism theorems in the symmetric case. Having developed some general material on Poisson processes in Sect. 7, we make use of it in the next two sections. Section 8 contains an excursion theory proof of the generalized second Ray-Knight theorem, and Sect. 9 explains a similar theorem for random interlacements. Up till this point proofs use the method of moments. In Sect. 10 we explain how to prove these theorems using the method of Laplace transforms.