Analysis and geometry on marked configuration spaces

Abstract

We carry out analysis and geometry on a marked configuration space $Ω^M_X$ over a Riemannian manifold $X$ with marks from a space $M$. We suppose that $M$ is a homogeneous space $M$ of a Lie group $G$. As a transformation group $\frak A$ on $Ω_X^M$ we take the ``lifting'' to $Ω_X^M$ of the action on $X\times M$ of the semidirect product of the group $\operatorname{Diff}_0(X)$ of diffeomorphisms on $X$ with compact support and the group $G^X$ of smooth currents, i.e., all $C^\infty$ mappings of $X$ into $G$ which are equal to the identity element outside of a compact set. The marked Poisson measure $π_σ$ on $Ω_X^M$ with Lévy measure $σ$ on $X\times M$ is proven to be quasiinvariant under the action of $\frak A$. Then, we derive a geometry on $Ω_X^M$ by a natural ``lifting'' of the corresponding geometry on $X\times M$. In particular, we construct a gradient $\nabla^Ω$ and a divergence $\operatorname{div}^Ω$. The associated volume elements, i.e., all probability measures $μ$ on $Ω_X^M$ with respect to which $\nabla^Ω$ and $\operatorname{div}^Ω$ become dual operators on $L^2(Ω_X^M;μ)$, are identified as the mixed marked Poisson measures with mean measure equal to a multiple of $σ$. As a direct consequence of our results, we obtain marked Poisson space representations of the group $\frak A$ and its Lie algebra $\frak a$. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures.