PUB - Publikationen an der Universität Bielefeld

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(-tHGamma))(tis an element ofR+) was introduced and studied in [J. Funct. Anal. 154 (1998), 444-500]. Here, H-Gamma is the Dirichlet operator of the Dirichlet form epsilonGamma 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(infinity) that is continuously embedded into Gamma(X). Under some conditions on the manifold X, we prove that Gamma(infinity) is a set of full pi(m) measure and derive an explicit formula for the heat semigroup: (e(-tHGamma) F) (gamma) = integral(Gammainfinity) F(xi)P-t,P-gamma(dxi), where P-t,P-gamma is a probability measure on Gamma(infinity) for all t > 0, gamma is an element of Gamma(infinity). 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(infinity). The second one, obtained in the case X = R-d, is the Feller property with respect to the intrinsic metric of the Dirichlet form epsilon(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 is an element of Gammainfinity will never leave Gamma(infinity) and has continuous sample path in Gamma(infinity) provided dim X greater than or equal to 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the P-t,P-gamma((.)) 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(infinity) from the underlying Brownian motion.