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45
Diffeomorphism Groups And Current Algebras: configuration space analysis in quantum theory
 PREPRINT 97073, SFB 343
, 1997
"... The constuction of models of nonrelativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the fr ..."
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Cited by 17 (1 self)
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The constuction of models of nonrelativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on R^d.
Glauber dynamics of continuous particle systems
"... This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ cor ..."
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Cited by 16 (7 self)
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This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ,µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ: = ∫ Rd(1 − e−φ(x))z dx < 1, we also prove that the generator H has a spectral gap ≥ 1−δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6].
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
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...
Schied: Rademacher’s theorem on configuration spaces and applications
, 1998
"... Abstract: We consider an L 2Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρLipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular ..."
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Cited by 8 (3 self)
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Abstract: We consider an L 2Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρLipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular fulfilled by the classical Poisson measures and by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of ρ for a set to be exceptional. This result immediately implies, for instance, a quasisure version of the spatial ergodic theorem. We also show that ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form. 0. Introduction. Let ΓX be the configuration space over a Riemannian manifold X. In this paper, we consider a class of probability measures on ΓX, which in particular contains certain Ruelle type Gibbs measures and mixed Poisson measures. Using a natural ‘nonflat ’ geometric structure of ΓX, recently analyzed in Albeverio, Kondratiev and Röckner (1996a),
Explicit stochastic analysis of Brownian motion and point measures on Riemannian manifolds
 J. Funct. Anal
, 1999
"... The gradient and divergence operators of stochastic analysis on Riemannian manifolds are expressed using the gradient and divergence of the flat Brownian motion. By this method we obtain the almostsure version of several useful identities that are usually stated under expectations. The manifoldvalu ..."
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Cited by 7 (3 self)
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The gradient and divergence operators of stochastic analysis on Riemannian manifolds are expressed using the gradient and divergence of the flat Brownian motion. By this method we obtain the almostsure version of several useful identities that are usually stated under expectations. The manifoldvalued Brownian motion and random point measures on manifolds are treated successively in the same framework, and stochastic analysis of the Brownian motion on a Riemannian manifold turns out to be closely related to classical stochastic calculus for jump processes. In the setting of point measures we introduce a damped gradient that was lacking in the multidimensional case. Key words: Stochastic calculus of variations, Brownian motion, random measures, Riemannian manifolds. Mathematics Subject Classification (1991). 60H07, 60H25, 5899, 58C20, 58G32, 58G99. 1 Introduction The IR d valued Brownian motion on the Wiener space (W; F W ; ) gathers many properties that are important in stoch...
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
Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density
, 2002
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On a relation between intrinsic and extrinsic Dirichlet forms for interacting particle systems
, 1998
"... In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic preDirichlet form E \Gamma ß oe of the Poisson measure ß oe in terms of the extrinsic one E \Gamma ß oe ;H X oe . More precisely, replacing ß oe by a Gibbs measure ¯ on the confi ..."
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Cited by 6 (0 self)
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In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic preDirichlet form E \Gamma ß oe of the Poisson measure ß oe in terms of the extrinsic one E \Gamma ß oe ;H X oe . More precisely, replacing ß oe by a Gibbs measure ¯ on the configuration space \Gamma X we derive a relation between the intrinsic preDirichlet form E \Gamma ¯ of the measure ¯ and the extrinsic one E P ¯;H X oe . As a consequence we prove the closability of E \Gamma ¯ on L 2 (\Gamma X ; ¯) under very general assumptions on the interaction potential of the Gibbs measures µ.