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Laplace operators on differential forms over configuration spaces
 J. Geom. Phys
"... Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given. 2000 AMS Mathematics Subject Classification. 60G57, ..."
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Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given. 2000 AMS Mathematics Subject Classification. 60G57, 58A10Contents
de Rham cohomology of configuration spaces with Poisson measure
 J. Funct. Anal
, 1995
"... The space ΓX of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of squareintegrable differential forms over ΓX, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitari ..."
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Cited by 3 (0 self)
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The space ΓX of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of squareintegrable differential forms over ΓX, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the L 2cohomology of the underlying manifold X.
ANALYSIS AND GEOMETRY ON MARKED CONFIGURATION SPACES
, 2006
"... We carry out analysis and geometry on a marked configuration space ΩM X over a Riemannian manifold X with marks from a space M. We suppose that M is a homogeneous space M of a Lie group G. As a transformation group A on ΩM X we take the “lifting ” to ΩMX of the action on X×M of the semidirect produc ..."
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We carry out analysis and geometry on a marked configuration space ΩM X over a Riemannian manifold X with marks from a space M. We suppose that M is a homogeneous space M of a Lie group G. As a transformation group A on ΩM X we take the “lifting ” to ΩMX of the action on X×M of the semidirect product of the group Diff0(X) of diffeomorphisms on X with compact support and the group GX of smooth currents, i.e., all C ∞ mappings of X into G which are equal to the identity element outside of a compact set. The marked Poisson measure πσ on ΩM X with Lévy measure σ on X × M is proven to be quasiinvariant under the action of A. Then, we derive a geometry on by a natural “lifting ” of the corresponding geometry on X × M. In particular, we construct a ΩM X gradient ∇Ω and a divergence div Ω. The associated volume elements, i.e., all probability measures µ on ΩM X with respect to which ∇Ω and div Ω become dual operators on L2 (ΩM X; µ), are identified as the mixed marked Poisson measures with mean measure equal to a multiple of σ. As a direct consequence of our results, we obtain marked Poisson space representations of the group A and its Lie algebra a. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures. 1991 AMS Mathematics Subject Classification. Primary 60G57. Secondary 57S10, 54H15 0