Weitzenböck Formulas on Poisson Probability Spaces

This paper surveys and compares some recent approaches to stochastic infinite-dimensional geometry on the space Γ of configurations (i. e. locally finite subsets) of a Riemannian manifold M under Poisson measures. In particular, different approaches to Bochner– Weitzenböck formulas are considered. A unitary transform is also introduced by mapping functions of n configuration points to their multiple stochastic integral. 1. Weitzenböck Formula under a Measure LetM be a Riemannian manifold with volume measure dx, covariant derivative ∇, and exterior derivative d. Let ∇∗ μ and dμ denote the adjoints of ∇ and d under a measure μ on M of the form μ( dx) = e dx. The classical Weitzenböck formula under the measure μ states that dμ d + dd ∗ μ = ∇∗ μ∇+R−Hessφ , where R denotes the Ricci tensor on M . In terms of the de Rham Laplacian HR = dμ d + dd ∗ μ and of the Bochner Laplacian HB = ∇∗ μ∇ we have HR = HB +R−Hessφ . In particular the term Hessφ plays the role of a curvature under the measure μ. ∗ Permanent address: Université de La Rochelle, 17042 La Rochelle, France.