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Analysis and Geometry on Configuration Spaces
, 1997
"... In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X ..."
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Cited by 50 (8 self)
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In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...
Analysis and geometry on configuration spaces: The Gibbsian case
, 1998
"... Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is th ..."
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Cited by 10 (2 self)
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Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is the gradient of the Riemannian structure on \Gamma one can define a corresponding divergence div \Gamma OE such that the canonical Gibbs measures are exactly those measures ¯ for which r \Gamma and div \Gamma OE are dual operators on L 2 (\Gamma; ¯). One consequence is that for such ¯ the corresponding Dirichlet forms E \Gamma ¯ are defined. In addition, each of them is shown to be associated with a conservative diffusion process on \Gamma with invariant measure ¯. The corresponding generators are extensions of the operator \Delta \Gamma OE := div \Gamma OE r \Gamma . The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brown...
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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Cited by 2 (0 self)
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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