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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].
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 6 (4 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.
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
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 5 (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.
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).
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 4 (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.
Application of the lent particle method to Poisson driven SDE’s", Probability Theory and Related Fields 151
, 2011
"... We apply the Dirichlet forms version of Malliavin calculus to stochastic differential equations with jumps. As in the continuous case this weakens significantly the assumptions on the coefficients of the SDE. In spite of the use of the Dirichlet forms theory, this approach brings also an important s ..."
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Cited by 4 (4 self)
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We apply the Dirichlet forms version of Malliavin calculus to stochastic differential equations with jumps. As in the continuous case this weakens significantly the assumptions on the coefficients of the SDE. In spite of the use of the Dirichlet forms theory, this approach brings also an important simplification which was not available nor visible previously: an explicit formula giving the carré du champ matrix, i.e. the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples. In this paper which uses the construction done in [7] we restrict ourselves to the existence of densities, smoothness will be studied separately.
M.: Equilibrium Kawasaki dynamics of continuous particle systems
 Infin. Dimens. Anal. Quantum Probab. Relat. Top
, 2007
"... We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a prior ..."
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Cited by 3 (2 self)
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We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birthanddeath process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).
The Law of Large Numbers and the Law of the Iterated Logarithm for Infinite Dimensional Interacting Diffusion Processes
, 2000
"... The classical Dirichlet form given by the intrinsic gradient on # R d is associated with a Markov process consisting of a countable family of interacting di#usions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure ..."
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Cited by 2 (1 self)
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The classical Dirichlet form given by the intrinsic gradient on # R d is associated with a Markov process consisting of a countable family of interacting di#usions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasisure analysis of Dirichlet forms is used to find exceptional sets of configurations for this Markov process. We consider large scale properties of the configuration and show that, for quite general measures, the process never hits those unusual configurations that violate the law of large numbers. Furthermore, for certain Gibbs measures, which model random particles in R d that interact via a potential function, we show, for d # 3, that the process never hits those unusual configurations that violate the law of the iterated logarithm.