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 $(e^{-tH^\Gamma})_{t\in\R_+}$ was introduced and studied in [{\it J. Func. Anal.} {\bf 154} (1998), 444--500]. Here, $H^\Gamma$ is the Dirichlet operator of the Dirichlet form ${\cal E}^\Gamma$ 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 $X$ and we prove that $\Gamma_\infty$ is a set of full $\pi_m$ measure. The central results of the paper are two types of Feller properties for the heat semigroup. 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 $\operatorname{dim}X\ge2$. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the $\P_{t,\gamma}(\cdot)$ above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case $\operatorname{dim}X=1$. Finally, as an easy consequence we get a ``path-wise'' construction of the independent particle process on $\Gamma_\infty$ from the underlying Brownian motion.