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Infinite interaction diffusion particles I: Equilibrium process and its scaling limit
 Forum Math
"... A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y) ..."
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Cited by 4 (3 self)
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A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics—the socalled gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form EΓ µ on L2 (Γ; µ), and under general conditions on the potential φ, prove its closability. For a potential φ having a “weak ” singularity at zero, we also write down an explicit form of the generator of EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′), where D ′ is the dual space of D:=C ∞ 0 (Rd).
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 4 (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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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
STOCHASTIC EQUIVARIANT COHOMOLOGIES AND CYCLIC COHOMOLOGY
, 2005
"... We give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the Chen–Souriau sense and to establish a link with cyclic cohomology. With the second one, we can establish a stochastic fixed point theorem. 1. Introduction. Let ..."
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We give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the Chen–Souriau sense and to establish a link with cyclic cohomology. With the second one, we can establish a stochastic fixed point theorem. 1. Introduction. Let
Laplace operators in deRham complexes associated with measures on configuration spaces
, 2001
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harmonic forms
"... Let X be a Riemannian manifold endowed with a cocompact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vectorvalued forms on the product manifold X N, which are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions ( ..."
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Let X be a Riemannian manifold endowed with a cocompact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vectorvalued forms on the product manifold X N, which are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions (the braided L 2 Betti numbers).