The Heat Semigroup on Configuration Spaces

Abstract

In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural “Riemannian-like” structure of the configuration space ${\Gamma }_X$ over a complete, connected, oriented, and stochastically complete Riemannian manifold $X$ of infinite volume, the heat semigroup $(e^{-t{H}^{\Gamma }}{)}_{t\in R_{+}}$ was introduced and studied in [J. Funct. Anal. 154 (1998), 444-500]. Here, $H^{\Gamma }$ is the Dirichlet operator of the Dirichlet form ${\mathcal{E}}^{\Gamma }$ over the space $L^2({\Gamma }_X,{\pi }_m)$, where ${\pi }_m$ is the Poisson measure on ${\Gamma }_X$ with intensity $m$—the volume measure on $X$. We construct a metric space ${\Gamma }_{\infty }$ that is continuously embedded into ${\Gamma }_X$. Under some conditions on the manifold $X$, we prove that ${\Gamma }_{\infty }$ is a set of full ${\pi }_m$ measure and derive an explicit formula for the heat semigroup: $(e^{-t{H}^{\Gamma }}F)(\gamma )={\int }_{{\Gamma }_{\infty }}F(\xi ){\mathbf{P}}_{t,\gamma }(d\xi )$, where ${\mathbf{P}}_{t,\gamma }$ is a probability measure on ${\Gamma }_{\infty }$ for all $t>0$, $\gamma \in {\Gamma }_{\infty }$. The central results of the paper are two types of Feller properties for the heat semigroup. The first one is a kind of strong Feller property with respect to the metric on the space ${\Gamma }_{\infty }$. The second one, obtained in the case $X=R^d$, is the Feller property with respect to the intrinsic metric of the Dirichlet form ${\mathcal{E}}^{\Gamma }$. Next, we give a direct construction of the independent infinite particle process on the manifold $X$, which is a realization of the Brownian motion on the configuration space. The main point here is that we prove that this process can start in every $\gamma \in {\Gamma }_{\infty }$, will never leave ${\Gamma }_{\infty }$, and has continuous sample path in ${\Gamma }_{\infty }$, provided $\dim X\ge 2$. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the ${\mathbf{P}}_{t,\gamma }(\cdot )$ above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case $\dim X=1$. Finally, as an easy consequence we get a “path-wise” construction of the independent particle process on ${\Gamma }_{\infty }$ from the underlying Brownian motion.