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Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
Abstract

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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
Dirichlet forms: Some infinite dimensional examples
"... 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 di#usion of gas molecules or the genetic evolution of a population. Each such system is made up of an e#e ..."
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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 di#usion of gas molecules or the genetic evolution of a population. Each such system is made up of an e#ectively 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