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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.
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).
Laplace operators in deRham complexes associated with measures on configuration spaces
, 2001
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Department of Mathematics PREPRINT SERIES 2011/2012 NO: 5 TITLE: ‘CLUSTER POINT PROCESSES ON MANIFOLDS’
"... The probability distribution µcl of a general cluster point process in a Riemannian manifold X (with independent random clusters attached to points of a configuration with distribution µ) is studied via the projection of an auxiliary measure ˆµ in the space of configurations ˆγ = {(x, ¯y)} ⊂ X × X, ..."
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The probability distribution µcl of a general cluster point process in a Riemannian manifold X (with independent random clusters attached to points of a configuration with distribution µ) is studied via the projection of an auxiliary measure ˆµ in the space of configurations ˆγ = {(x, ¯y)} ⊂ X × X, where x ∈ X indicates a cluster “centre” and ¯y ∈ X: = ⊔ nXn represents a corresponding cluster relative to x. We show that the measure µcl is quasiinvariant with respect to the group Diff0(X) of compactly supported diffeomorphisms of X, and prove an integrationbyparts formula for µcl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. General constructions are illustrated by examples including Euclidean spaces, Lie groups, homogeneous spaces, Riemannian manifolds and metric spaces. The paper is an extension of our earlier results for Poisson cluster measures [J. Funct. Analysis 256 (2009) 432–478] and for Gibbs cluster measures