A support property for infinite-dimensional interacting diffusion processes

Röckner M, Schmuland B. A support property for infinite-dimensional interacting diffusion processes. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 1998;326(3):359-364.The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space Gamma = {Z(+)-valued Radon measures on R-d}. We show that under mild conditions, the set Gamma/Gamma is epsilon-exceptional, where Gamma is the space of locally finite configurations in R-d, that is, measures gamma is an element of Gamma satisfying sup(x is an element of Rd) gamma({x}) less than or equal to 1. Thus, the associated diffusion lives on the smaller space Gamma. This result also holds for Gibbs measures with superstable interactions