Potential Theory of Hyperfinite Dirichlet Forms

Abstract

Probabilistic potential theory has been a very important component in the study of hyperfinite Dirichlet space theory. It provides a probabilistic inter- pretation of potential theory; and, more generally, it establishes a beautiful bridge between functional analysis and the theory of Markov processes. There are many applications of this theory, especially in the area of infi- nite dimensional stochastic analysis and mathematical physics. Our purpose in this chapter is to develop the probabilistic potential theory associated with hyperfinite Dirichlet forms and the related Markov chains. The motivation is twofold. On the one hand, we want to establish a relationship between the standard Dirichlet space theory and the hyperfinite counterpart. On the other hand, we want to provide new methods for the theory of hyperfinite Dirichlet forms itself. Infinite dimensional stochastic analysis has been devel- oped extensively in the last decades. We hope to convince the reader that nonstandard analysis can provide a new tool to deal with problems in this exciting area, see particularly Chap. 4, for example.