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51
Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications
 J. Funct. Anal
, 1994
"... In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = ..."
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Cited by 34 (16 self)
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In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = \Delta H is the GrossLaplacian, (E; H; fl) is an abstract Wiener space and B = \Gammaid E +v where v takes values in the CameronMartin space H . Using Gross' logarithmic Sobolevinequality in an essential way we show that ¯ is always absolutely continuous w.r.t. the Gaussian measure fl and that the square root of the density is in the Malliavin test function space of order 1 in L 2 (fl). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L ¯ = 0. These include results on the "inverse problem", i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove ...
Quasiregular Dirichlet forms: Examples and counterexamples
, 1993
"... We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of exam ..."
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Cited by 20 (7 self)
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We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)elliptic part, diffusions on loops spaces, and certain FlemingViot processes.
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 10 (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).
Minoru W.: Multiple Stochastic Integrals Construction of nonGaussian Reflection Positive Generalized Random Fields
"... Multiple stochastic integrals construction of nonGaussian reflection positive generalized random fields S. Albeverio1),2), M. W. Yoshida3)∗ ..."
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Cited by 9 (8 self)
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Multiple stochastic integrals construction of nonGaussian reflection positive generalized random fields S. Albeverio1),2), M. W. Yoshida3)∗
Neerven, On closability of directional gradients
 Potential Anal
"... ABSTRACT. Let be a centred Gaussian measure on a separable real Banach space E, and let H be a Hilbert subspace of E. We provide necessary and sufcient conditions for closability in Lp(E;) of the gradient DH in the direction of H. These conditions are further elaborated in case when the gradient DH ..."
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Cited by 9 (2 self)
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ABSTRACT. Let be a centred Gaussian measure on a separable real Banach space E, and let H be a Hilbert subspace of E. We provide necessary and sufcient conditions for closability in Lp(E;) of the gradient DH in the direction of H. These conditions are further elaborated in case when the gradient DH corresponds to a bilinear form associated with a certain nonsymmetric OrnsteinUhlenbeck operator. Some natural examples of closability and nonclosability are presented. 1.
Energy image density property and the lent particle method for Poisson measures
 Jour. of Functional Analysis
"... We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the energy image density property for Dirichlet forms. The associated gradient is a local operator and gives rise to a nice formula called the lent particle method which consists in adding a particle an ..."
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Cited by 9 (6 self)
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We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the energy image density property for Dirichlet forms. The associated gradient is a local operator and gives rise to a nice formula called the lent particle method which consists in adding a particle and taking it back after some calculation.
Strong uniqueness for a class of infinite dimensional Dirichlet operators and applications to stochastic quantization
, 1997
"... Strong and Markov uniqueness problems in L² for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L^psetting is discussed. As a direct application essential self adjointness and strong uniqueness ..."
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Cited by 7 (1 self)
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Strong and Markov uniqueness problems in L² for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L^psetting is discussed. As a direct application essential self adjointness and strong uniqueness in L^p is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume ae R².
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 7 (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.