Lectures on Isomorphism Theorems

TL;DR

This paper provides an accessible overview of isomorphism theorems connecting Markov local times and Gaussian processes, highlighting recent developments and applications in graph cover times and non-symmetric processes.

Contribution

It offers a new, informal introduction to isomorphism theorems, including recent advances and their applications in various stochastic processes.

Findings

Connections between Markov local times and Gaussian processes clarified

Recent applications to graph cover times and Gaussian fields discussed

New isomorphism theorems for non-symmetric Markov processes introduced

Abstract

These notes originated in a series of lectures I gave in Marseille in May, 2013. I was invited to give an introduction to the isomorphism theorems, originating with Dynkin, which connect Markov local times and Gaussian processes. This is an area I had worked on some time ago, and even written a book about, but had then moved on to other things. However, isomorphism theorems have become of interest once again, both because of new applications to the study of cover times of graphs and Gaussian fields, and because of new isomorphism theorems for non-symmetric Markov processes and their connection with loop soups and Poisson processes. Thus I felt the time was ripe for a new introduction to this topic. In these notes I have tried to preserve the informal atmosphere of the lectures, and often simply refer the reader to my book and other sources for details.