The heat semigroup on configuration spaces
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In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural ``Riemannian-like'' structure of the configuration space $Γ_X$ over a complete, connected, oriented, and stochastically complete Riemannian manifold $X$ of infinite volume, the heat semigroup $(e^{-tH^Γ})_{t\in\R_+}$ was introduced and studied in [{\it J. Func. Anal.} {\bf 154} (1998), 444--500]. Here, $H^Γ$ is the Dirichlet operator of the Dirichlet form ${\cal E}^Γ$ over the space $L^2(Γ_X,π_m)$, where $π_m$ is the Poisson measure on $Γ_X$ with intensity $m$--the volume measure on $X$. We construct a metric space $Γ_\infty$ that is continuously embedded into $Γ_X$. Under some conditions on the manifold $X$ and we prove that $Γ_\infty$ is a set of full $π_m$ 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 $γ\inΓ_\infty$, will never leave $Γ_\infty$, and has continuous sample path in $Γ_\infty$, 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 $¶_{t,γ}(\cdot)$ 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 $Γ_\infty$ from the underlying Brownian motion.