Analysis and geometry on marked configuration spaces

Details Analysis and geometry on marked configuration spaces Analysis and geometry on marked configuration spaces

2006-08-14

arxiv.org/abs/math/0608344v1

Published in "Infinite Dimensional Harmonic Analysis (Kyoto, September 20-24, 1999)", (H. Heyer et al., eds), pp. 1-39, Gr\"ab

Mathematics

Probability

Scientific paper

We carry out analysis and geometry on a marked configuration space $\Omega^M_X$ 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 $\Omega_X^M$ we take the ``lifting'' to $\Omega_X^M$ 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 $\pi_\sigma$ on $\Omega_X^M$ with L\'evy measure $\sigma$ on $X\times M$ is proven to be quasiinvariant under the action of $\frak A$. Then, we derive a geometry on $\Omega_X^M$ by a natural ``lifting'' of the corresponding geometry on $X\times M$. In particular, we construct a gradient $\nabla^\Omega$ and a divergence $\operatorname{div}^\Omega$. The associated volume elements, i.e., all probability measures $\mu$ on $\Omega_X^M$ with respect to which $\nabla^\Omega$ and $\operatorname{div}^\Omega$ become dual operators on $L^2(\Omega_X^M;\mu)$, are identified as the mixed marked Poisson measures with mean measure equal to a multiple of $\sigma$. 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.

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