JournalsprimsVol. 39, No. 1pp. 1–48

The Heat Semigroup on Configuration Spaces

  • Michael Röckner

    Universität Bielefeld, Germany
  • Yuri Kondratiev

    Universität Bielefeld, Germany
  • Eugene Lytvynov

    Universität Bonn, Germany
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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) Pt,γ(dξ), where Pt,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 dim X ≥ 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the Pt,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