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 X over a complete, connected, oriented, and stochastically complete Rie- mannian manifold X of infinite volume, the heat semigroup (e −tH )t∈R+ was introduced and studied in (J. Func. Anal. 154 (1998), 444-500). Here, H is the Dirichlet operator of the Dirichlet form E over the space L 2 ( X,�m), wherem is the Poisson measure on X with intensity m—the volume mea- sure on X. We construct a metric space ∞ that is continuously embedded into X. Under some conditions on the manifold X, we prove that ∞ is a set of fullm measure and derive an explicit formula for the heat semigroup: (e −tH F)() = R ∞ F(�) Pt,(d�), where Pt, is a probability measure on ∞ for all t > 0, ∈ ∞. The central results of the paper are two types of Feller prop- erties for the heat semigroup. The first one is a kind of strong Feller property with respect to the metric on the space ∞. The second one, obtained in the case X = R d , is the Feller property with respect to the intrinsic metric of the Dirichlet form E. Next, we give a direct construction of the independent infi- nite 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 ∈ ∞, will never leave ∞, and has continuous sample path in ∞, provided dimX ≥ 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the Pt,(� ) above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case dimX = 1. Finally, as an easy consequence we get a path-wise construction of the independent particle process on ∞ from the underlying Brownian motion.