A support property for infinite dimensional interacting diffusion processes

The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space $\ddot\Gamma=$ $\{Z_+$-valued Radon measures on \IR^d\}. We show that under mild conditions, the set $\ddot\Gamma\setminus\Gamma$ is \e-exceptional, where $\Gamma$ is the space of locally finite configurations in \IR^d, that is, measures $\gamma\in\ddot\Gamma$ satisfying \sup_{x\in\IR^d}\gamma(\{x\})\leq 1. Thus, the associated diffusion lives on the smaller space $\Gamma$. This result also holds for Gibbs measures with superstable interactions.Comment: French title: Une propri\'et\'e de support pour des processus de diffusion en dimension infinie avec interactio

Some Exceptional Configurations

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..

On the Local Property for Positivity Preserving Coercive Forms

. We show that, under mild conditions, two well-known definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from di#erential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between two di#erent notions of locality that have appeared in the literature of Dirichlet forms. The first is a slightly modified version of the definition of locality found in the book of Bouleau and Hirsch [BH 91; Chapter I, Corollary 5.1.4], while the second comes from the book of Ma and Rockner [MR 92; Chapter V, Proposition 1.2]. But here we do not assume that the form satisfies any normal contraction property, but only that it is positivity preserving (see Definition 0.1 below). Let (E, F , m) be a measure space, and suppose (E , D(E)) is a densely defined, closed, bilinear form on L 2 (E, F , m). Following [MR 92], we call such a form (E , D(E)) coercive if E(u, u) # 0 for all u ..