Some Exceptional Configurations

Abstract

The Dirichlet form given by the intrinsic gradient on Poisson space is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasi-sure analysis of Dirichlet forms is used to find exceptional sets for this Markov process. We show that the process never hits certain unusual configurations, such as those with more than unit mass at some position, or those that violate the law of large numbers. Some of these results also hold for Gibbs measures with superstable interactions. AMS (1991) subject classification 60H07, 31C25, 60G57, 60G60 1 Introduction In recent work [1, 2, 3, 4, 5, 11, 12, 13, 15, 18] the theory of Dirichlet forms has been used to construct and study Markov processes that take values in the space \Gamma X of locally finite configurations on a Riemannian manifold X . The configurati..