Differential Geometry on Compound Poisson Space

Abstract

In this paper we carry out analysis and geometry for a class of infinite dimensional manifolds, namely, compound configuration spaces as a natural generalization of the work \cite{AKR97}. More precisely a differential geometry is constructed on the compound configuration space $\Omega_{X}$ over a Riemannian manifold $X$. This geometry is obtained as a natural lifting of the Riemannian structure on $X$. In particular, the intrinsic gradient, divergence, and Laplace-Beltrami operator are constructed. Therefore the corresponding Dirichlet forms on $L^{2}(\Omega_{X})$ can be defined. Each is shown to be associated with a diffusion process on $\Omega_{X}$ (so called equilibrium process) which is nothing but the diffusion process on the simple configuration space $\Gamma_{X}$ together with corresponding marks. As another consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields on $X$ on compound Poisson space. Finally generalizations to the case when the compound Poisson measure is replaced by a marked Poisson measure easily follow from this construction.