# The Heat Semigroup on Configuration Spaces

### Yuri Kondratiev

Universität Bielefeld, Germany### Eugene Lytvynov

Universität Bonn, Germany### Michael Röckner

Universität Bielefeld, 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: $(e_{−tH_{Γ}}F)(γ)=∫_{Γ_{∞}}F(ξ)P_{t,γ}(dξ)$, 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 $X=R_{d}$, 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 $X$, 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 $dimX≥2$. 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 $dimX=1$. 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

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

DOI 10.2977/PRIMS/1145476147