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21
Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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Cited by 26 (3 self)
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
Equivalence of Gradients on Configuration Spaces
 Random Operators and Stochastic Equations
, 1999
"... The gradient on a Riemannian manifold X is lifted to the configuration space \Upsilon X on X via a pointwise identity. This entails a norm equivalence that either holds under any probability measure or characterizes the Poisson measures, depending on the tangent space chosen on \Upsilon X . More ..."
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Cited by 14 (4 self)
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The gradient on a Riemannian manifold X is lifted to the configuration space \Upsilon X on X via a pointwise identity. This entails a norm equivalence that either holds under any probability measure or characterizes the Poisson measures, depending on the tangent space chosen on \Upsilon X . More generally, this approach links carr'e du champ operators on X to their counterparts on \Upsilon X , and also includes structures that do not admit a gradient. Key words: Configuration spaces, Poisson measures, Stochastic analysis. Mathematics Subject Classification (1991): 58G32, 60H07, 60J45, 60J75. 1 Introduction Stochastic analysis under Poisson measures, cf. [5], [6], has been developed in several different directions. This is mainly due to the fact that, unlike on the Wiener space, the gradient on Fock space and the infinitesimal Poisson gradient do not coincide under the identification of the Fock space to the L 2 space of the Poisson process.  The gradient on Fock space is in...
Connections and curvature in the Riemannian geometry of configuration spaces
, 2001
"... Torsion free connections and a notion of curvature are introduced on the infinite dimensional nonlinear configuration space \Gamma of a Riemannian manifold M under a Poisson measure. This allows to state identities of Weitzenbock type and energy identities for anticipating stochastic integral opera ..."
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Cited by 10 (1 self)
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Torsion free connections and a notion of curvature are introduced on the infinite dimensional nonlinear configuration space \Gamma of a Riemannian manifold M under a Poisson measure. This allows to state identities of Weitzenbock type and energy identities for anticipating stochastic integral operators. The onedimensional Poisson case itself gives rise to a nontrivial geometry, a de RhamHodge Kodaira operator, and a notion of Ricci tensor under the Poisson measure. The methods used in this paper have been so far applied to ddimensional Brownian path groups, and rely on the introduction of a particular tangent bundle and associated damped gradient. Key words: Configuration spaces, Poisson spaces, covariant derivatives, curvature, connections. Mathematics Subject Classification (1991). Primary: 60H07, 58G32, 53B21. Secondary: 53B05, 58A10, 58C35, 60H25. 1
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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Cited by 7 (3 self)
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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
Scaling limit of stochastic dynamics in classical continuous systems
, 2002
"... We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on R d, d ≥ 1. The aim is to derive macroscopic quantities from a given micro or mesoscopic system. The scaling we ..."
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Cited by 5 (5 self)
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We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on R d, d ≥ 1. The aim is to derive macroscopic quantities from a given micro or mesoscopic system. The scaling we
Laplace operators and diffusions in tangent bundles over Poisson spaces
 Preprint SFB 256 No. 629, Universität
, 1999
"... Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1forms and associated semigroups are considered. Their probabilistic interpretation is given. 1 ..."
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Cited by 5 (4 self)
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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 1forms and associated semigroups are considered. Their probabilistic interpretation is given. 1
Marked Gibbs measures via cluster expansion
, 1998
"... We give a sufficiently detailed account on the construction of marked Gibbs measures in the high temperature and low fugacity regime. This is proved for a wide class of underlying spaces and potentials such that stability and integrability conditions are satisfied. That is, for state space we take a ..."
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Cited by 4 (0 self)
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We give a sufficiently detailed account on the construction of marked Gibbs measures in the high temperature and low fugacity regime. This is proved for a wide class of underlying spaces and potentials such that stability and integrability conditions are satisfied. That is, for state space we take a locally compact separable metric space X and a separable metric space S for the mark space. This framework allowed us to cover several models of classical and quantum statistical physics. Furthermore, we also show how to extend the construction for more general spaces as e.g., separable standard Borel spaces. The construction of the marked Gibbs measures is based on the method of cluster expansion.
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).
Analysis on Configuration Spaces and Gibbs Cluster Ensembles
, 2007
"... The distribution µ of a Gibbs cluster point process in X = R d (with npoint clusters) is studied via the projection of an auxiliary Gibbs measure defined on the space of configurations in X × X n. We show that µ is quasiinvariant with respect to the group Diff0(X) of compactly supported diffeomorp ..."
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Cited by 4 (0 self)
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The distribution µ of a Gibbs cluster point process in X = R d (with npoint clusters) is studied via the projection of an auxiliary Gibbs measure defined on the space of configurations in X × X n. We show that µ is quasiinvariant with respect to the group Diff0(X) of compactly supported diffeomorphisms of X and prove an integrationbyparts formula for µ. The corresponding equilibrium stochastic dynamics is then constructed by using the method of Dirichlet forms.
Equilibrium Stochastic Dynamics of Poisson Cluster Ensembles
"... The distribution µ of a Poisson cluster process in X = R d (with npoint clusters) is studied via the projection of an auxiliary Poisson measure in the space of configurations in X n, with the intensity measure being the convolution of the background intensity (of cluster centres) with the probabili ..."
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Cited by 3 (1 self)
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The distribution µ of a Poisson cluster process in X = R d (with npoint clusters) is studied via the projection of an auxiliary Poisson measure in the space of configurations in X n, with the intensity measure being the convolution of the background intensity (of cluster centres) with the probability distribution of a generic cluster. We show that µ is quasiinvariant with respect to the group of compactly supported diffeomorphisms of X, and prove an integration by parts formula for µ. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.