Differential Geometry on Compound Poisson Space

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 [AKR98a]. More precisely a differential geometry is constructed on the compound configuration space ΩX over a Riemannian manifold X. This geometry is obtained as a natural lifting of the Riemannian structure on X. In particular, the intrinsic gradient ∇ΩX divergence divX πτ σ , and Laplace-Beltrami operator HX πτ σ = −div ΩX πτ σ ∇ΩX are constructed. Therefore the corresponding Dirichlet forms EX πτ σ on L (ΩX , π τ σ) can be defined. Each is shown to be associated with a diffusion process on ΩX (so called equilibrium process) which is nothing but the diffusion process on the simple configuration space ΓX together with corresponding marks, i.e, (Xω t ,mω). 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 πτ σ is replaced by a marked Poisson measure μσ⊗τ easily follow from this construction.