Introduction to Potential Theory

Abstract

One of the fruitful achievements of probability theory in recent years has been the recognition that two seemingly unrelated theories in physics—one for Brownian motion and one for potentials—are mathematically equivalent. That is, the results of the two theories are in one-to-one correspondence and any proof of a result in one theory can be translated directly into a proof of the corresponding result in the other theory. In this chapter we shall sketch how this equivalence comes about, and we shall see that Brownian motion is a process which is like a Markov chain except that it does not have a denumerable state space and time does not proceed in discrete steps. The details of this equivalence can be found in Knapp [1965]. The important thing to notice will be that the definitions of potential-theoretic concepts in terms of Brownian motion do not depend on isolated specific properties of the process but depend only on the Markovian character of Brownian motion. In other words, there is reasonable hope of defining for an arbitrary Markov chain a potential theory in which analogs of the classical theorems hold.