Dirichlet forms: Some infinite dimensional examples

The theory of Dirichlet forms deserves to be better known. It is an area of Markov process theory that uses the energy of functionals to study a Markov process from a quantitative point of view. For instance, the recent notes of Saloff-Coste [S-C] use Dirichlet forms to analyze Markov chains with finite state space, by making energy comparisons. In this way, information about a simple chain is parlayed into information about another, more complicated chain. The upcoming book [AF] by Aldous and Fill will use Dirichlet forms for similar purposes. Dirichlet form theory does not use the tools of partial differential equations, as in standard diffusion theory, and therefore is not as closely tied to analysis on Euclidean space. For example, Dirichlet forms can be used to study Markov processes taking values in spaces of fractional dimension, i.e. fractals (see [F3], [Ku2], [KuY]). This paper applies Dirichlet form techniques to study Markov processes taking values in infinite dimensional spaces. Such processes are used to describe a complex natural phenomenon, such as the diffusion of gas molecules or the genetic evolution of a population. Each such system is made up of an effectively infinite number of individuals whose evolution in time is governed by a combination of random chance and interactions with the other individuals in the system. The complexity of such a system makes this a forbidding mathematical problem. This paper is not an introduction to Dirichlet form theory. We are not interested here in all the details and generalities of the theory; there are several good sources for that ([MR1] [BH] [FOT]). In fact, we do not even define Dirichlet forms, we simply motivate them. This paper is about calculations, and how you use energy estimates to give concrete results on the sample path properties of Markov processes. The four processes that we consider are: 1. Brownian motion on R. 2. Ornstein-Uhlenbeck process on Hilbert space. 3. Fleming-Viot process on a space of probability measures. 4. A particle process on configuration space.