# The Heat Semigroup on Configuration Spaces

### Michael Röckner

Universität Bielefeld, Germany### Yuri Kondratiev

Universität Bielefeld, Germany### Eugene Lytvynov

Universität Bonn, Germany

## 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 Γ_X_ over a complete, connected, oriented, and stochastically complete Riemannian manifold *X* of infinite volume, the heat semigroup (*e*-*tH_Γ) t_∈ℝ+ was introduced and studied in [J. Funct. Anal. 154 (1998), 444-500]. Here, H_Γ is the Dirichlet operator of the Dirichlet form 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∞ that is continuously embedded into Γ_X*. Under some conditions on the manifold

*X*, we prove that Γ∞ is a set of full π_m_ measure and derive an explicit formula for the heat semigroup: (

*e*-

*tH_Γ F)(γ) = ∫Γ∞ F(ξ)*, is the Feller property with respect to the intrinsic metric of the Dirichlet form _E_Γ. Next, we give a direct construction of the independent infinite particle process on the manifold

**P***t*,γ(dξ), where**P***t*,γ is a probability measure on Γ∞ for all*t*> 0, γ ∈ Γ∞. 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 Γ∞. The second one, obtained in the case*X*=ℝ_d*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 ∈ Γ∞, will never leave Γ∞, and has continuous sample path in Γ∞, provided dim

*X*≥ 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the

**P**

*t*,γ(•) 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 Γ∞ from the underlying Brownian motion.

## Cite this article

Michael Röckner, Yuri Kondratiev, Eugene Lytvynov, The Heat Semigroup on Configuration Spaces. Publ. Res. Inst. Math. Sci. 39 (2003), no. 1, pp. 1–48

DOI 10.2977/PRIMS/1145476147