A support property for infinite dimensional interacting diffusion processes /
Une propriété de support pour des processus de diffusion
en dimension infinie avec interaction.
Michael Röckner and Byron Schmuland
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 ${I\mskip-6mu R}^d\}$. We show that under mild conditions, the set $\ddot\Gamma\setminus\Gamma$ is $\mathcal E$-exceptional, where $\Gamma$ is the space of locally finite configurations in ${I\mskip-6mu R}^d$, that is, measures $\gamma\in\ddot\Gamma$ satisfying $\sup_{x\in{I\mskip-6mu R}^d}\gamma(\{x\})\leq 1$. Thus, the associated diffusion lives on the smaller space $\Gamma$. This result also holds for Gibbs measures with superstable interactions.