Differential Geometry on Compound Poisson Space

Details Differential Geometry on Compound Poisson Space Differential Geometry on Compound Poisson Space

1999-08-13

arxiv.org/abs/math/9908059v1

Methods of Functional Analysis and Topology, 4(1), pp. 32--58, 1998

Mathematics

Functional Analysis

44 pages, 2 Diagrams

Scientific paper

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.

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