Analysis and geometry on $R_+$-marked configuration spaces

Abstract

We carry out analysis and geometry on a marked configuration space $\Omega_X^{R_+}$ over a Riemannian manifold $X$ with marks from the space $R_+$ as a natural generalization of the work {\bf [}{\it J. Func. Anal}. {\bf 154} (1998), 444--500{\bf ]}. As a transformation group $\mathfrak G$ on this space, we take the ``lifting'' to $\Omega_X^{R_+}$ of the action on $X\times R_+$ of the semidirect product of the group Diff of diffeomorphisms on $X$ with compact support and the group $R_+^X$ of smooth currents, i.e., all $C^\infty$ mappings of $X$ into $R_+$ which are equal to one outside a compact set. The marked Poisson measure $\pi$ on $\Omega_X^{R_+}$ with L\'evy measure $\sigma$ is proven to be quasiinvariant under the action of $\mathfrak G$. Then, we derive a geometry on $\Omega_X^{R_+}$ by a natural ``lifting'' of the corresponding geometry on $X\times R_+$. In particular, we construct a gradient $\nabla^\Omega$ and divergence $div^\Omega$. The associated volume elements, i.e., all probability measures $\mu$ on $\Omega_X^{R_+}$ with respect to which $\nabla^\Omega$ and $div^\Omega$ become dual operators on $L^2(\Omega_X^{R_+} ,\mu)$ are identified as the mixed 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 $\mathfrak G$ and its Lie algebra $\mathfrak g$. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures. In particular, we obtain conditions of ergodicity of the semigroups generated by the Dirichlet operators. A possible generalization of the results of the paper to the case where the marks belong to a homogeneous space of a Lie group is noted.