Results 11  20
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21
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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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.
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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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.
ANALYSIS AND GEOMETRY ON MARKED CONFIGURATION SPACES
, 2006
"... We carry out analysis and geometry on a marked configuration space ΩM X over a Riemannian manifold X with marks from a space M. We suppose that M is a homogeneous space M of a Lie group G. As a transformation group A on ΩM X we take the “lifting ” to ΩMX of the action on X×M of the semidirect produc ..."
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We carry out analysis and geometry on a marked configuration space ΩM X over a Riemannian manifold X with marks from a space M. We suppose that M is a homogeneous space M of a Lie group G. As a transformation group A on ΩM X we take the “lifting ” to ΩMX of the action on X×M of the semidirect product of the group Diff0(X) of diffeomorphisms on X with compact support and the group GX of smooth currents, i.e., all C ∞ mappings of X into G which are equal to the identity element outside of a compact set. The marked Poisson measure πσ on ΩM X with Lévy measure σ on X × M is proven to be quasiinvariant under the action of A. Then, we derive a geometry on by a natural “lifting ” of the corresponding geometry on X × M. In particular, we construct a ΩM X gradient ∇Ω and a divergence div Ω. The associated volume elements, i.e., all probability measures µ on ΩM X with respect to which ∇Ω and div Ω become dual operators on L2 (ΩM X; µ), are identified as the mixed marked Poisson measures with mean measure equal to a multiple of σ. As a direct consequence of our results, we obtain marked Poisson space representations of the group A and its Lie algebra a. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures. 1991 AMS Mathematics Subject Classification. Primary 60G57. Secondary 57S10, 54H15 0
The maximum Markovian selfadjoint extensions of Dirichlet operators for interacting particle systems
"... Let µ be a Ruelle measure on the configuration space # R d with a pair potential #. Then the generator of the corresponding intrinsic Dirichlet form (E is an extension of the Dirichlet operator # # b defined by = div is the set of smooth cylinderfunder(6G Riemannianstrunnia ..."
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Let µ be a Ruelle measure on the configuration space # R d with a pair potential #. Then the generator of the corresponding intrinsic Dirichlet form (E is an extension of the Dirichlet operator # # b defined by = div is the set of smooth cylinderfunder(6G Riemannianstrunnia on # R d and div # the corresponding divergence. For a large class of nonnegative (singuj08 potentials #, we give a convergence characterization for the weak Sobolev spaces W (# R d ; ) and prove that the generator of (E (# R d ; )) is the maximu m Markovian selfadjoint extension of (# ). FuG66C(j00C we construj stationary di#utio processes associated with (E (# R d ; )) by approximation.
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"... Stock market trading rule discovery using technical charting heuristics ..."
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Stock market trading rule discovery using technical charting heuristics
Dirichlet forms: Some infinitedimensional examples
, 1999
"... The author presents three examples of a Markov process taking values in an infinitedimensional state space, and analyzes the sample path behaviour using the theory of Dirichlet forms. R ESUM E L'auteur presente trois exemples de processus de Markov a valeurs dans un espace d'etats de dimension infin ..."
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The author presents three examples of a Markov process taking values in an infinitedimensional state space, and analyzes the sample path behaviour using the theory of Dirichlet forms. R ESUM E L'auteur presente trois exemples de processus de Markov a valeurs dans un espace d'etats de dimension infinie et il analyse le comportement de leurs trajectoires au moyen de la theorie des formes de Dirichlet. 1. INTRODUCTION The theory of Dirichlet forms deserves to be better known. It is an area of Markov process theory that uses the energy of functionals to study a Markov process from a quantitative point of view. Recently, for instance, Salo#Coste (1997) used Dirichlet forms to analyze Markov chains with finite state spaces, by making energy comparisons. In this way, information about a simple chain is parlayed into information about another, more complicated chain. The upcoming book by Aldous and Fill (1999) will use Dirichlet forms for similar purposes. Dirichlet form theory does not use t...
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
harmonic forms
"... Let X be a Riemannian manifold endowed with a cocompact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vectorvalued forms on the product manifold X N, which are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions ( ..."
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Let X be a Riemannian manifold endowed with a cocompact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vectorvalued forms on the product manifold X N, which are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions (the braided L 2 Betti numbers).
Poisson cluster measures: quasiinvariance,
, 803
"... integration by parts and equilibrium stochastic dynamics ..."