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36
Weak Poincaré inequalities and L2convergence rates of Markov semigroups
 J. Funct. Anal
"... In order to describe L2convergence rates slower than exponential, the weak Poincare ́ inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare ́ inequality can be determined by each other. Conditions for the weak Poincare ́ inequality ..."
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Cited by 46 (8 self)
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In order to describe L2convergence rates slower than exponential, the weak Poincare ́ inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare ́ inequality can be determined by each other. Conditions for the weak Poincare ́ inequality to hold are presented, which are easy to check and which hold in many applications. The weak Poincaré inequality is also studied by using isoperimetric inequalities for diffusion and jump processes. Some typical examples are given to illustrate the general results. In particular, our results are applied to the stochastic quantization of field theory in finite volume. Moreover, a sharp criterion of weak Poincare ́ inequalities is presented for Poisson measures on configuration spaces.
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 38 (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
"... Abstract This Note aims to provide a short and selfcontained proof of an equivalence between square field operators on configuration spaces. A recent result in analysis on configuration spaces, namely an equivalence between Dirichlet forms, is retrieved as a particular case. Our method relies on d ..."
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Cited by 18 (5 self)
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Abstract This Note aims to provide a short and selfcontained proof of an equivalence between square field operators on configuration spaces. A recent result in analysis on configuration spaces, namely an equivalence between Dirichlet forms, is retrieved as a particular case. Our method relies on duality formulas and on extensions of the stochastic integral on Poisson space. Une équivalence de gradients ponctuelle sur les espaces de configurations Résume ́ Cette Note donne une preuve concise d’une équivalence entre opérateurs carre ́ du champ sur l’espace des configurations. Ceci permet de retrouver un résultat récent relatif a ̀ une équivalence entre formes de Dirichlet sur ce même espace. La méthode utilisée repose sur des formules de dualite ́ et sur les extensions de l’intégrale stochastique sur l’espace de Poisson. Version française abrégée L’analyse sur les espaces de configurations développée dans [1] repose sur une égalite ́ de normes en espérance pour le gradient local ∇Υ et l’opérateur de différence finie D sur les espaces des configurations ΥX sur une variéte ́ Riemannienne X. Dans cette Note nous présentons une version ponctuelle de cette relation, avec une preuve concise qui fait apparâıtre le rôle de la dualite ́ sur l’espace de Poisson. On montre que (∇ΥF,∇ΥG)L2γ(TX) = δ− (∇XDF,∇XDG)TX
Alexander: Traces of Semigroups Associated with Interacting Particle Systems
"... gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2006 Traces of semigroups associated with interacting particle systems ..."
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Cited by 17 (3 self)
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gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2006 Traces of semigroups associated with interacting particle systems
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 16 (2 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
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 13 (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 11 (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).
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 10 (5 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
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 8 (3 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.