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12
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...
Intrinsic metrics for nonlocal symmetric Dirichlet forms and applications to spectral theory. to appear
, 2010
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A dual characterization of length spaces with application to Dirichlet metric spaces
, 2009
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Wasserstein distance on configurations space
 Potential Anal
"... Abstract. In this paper, we provide upper bounds on several Rubinsteintype distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two wellknown gradients, in the sense of Malliavin calculus, which can be defined on this space. Actually, we show that de ..."
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Cited by 9 (2 self)
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Abstract. In this paper, we provide upper bounds on several Rubinsteintype distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two wellknown gradients, in the sense of Malliavin calculus, which can be defined on this space. Actually, we show that depending on the distance between configurations which is considered, it is one gradient or the other which is the most effective. Some applications to distance estimates between Poisson and other more sophisticated processes are also provided, and an investigation of our results to functional inequalities completes this work. 1.
Upper bounds on Rubinstein distances on configuration spaces and applications
 Communications on Stochastic Analysis
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The heat semigroup on configuration spaces
, 1992
"... In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural “Riemannianlike ” structure of the configuration space ΓX over a complete, connected, oriented, and stochastically complete Riemannian manifold X of infinite volume, the heat semigroup (e−tHΓ)t ..."
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Cited by 4 (1 self)
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In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural “Riemannianlike ” structure of the configuration space ΓX over a complete, connected, oriented, and stochastically complete Riemannian manifold X of infinite volume, the heat semigroup (e−tHΓ)t∈R+ was introduced and studied in [J. Func. Anal. 154 (1998), 444–500]. Here, H Γ is the Dirichlet operator of the Dirichlet form E Γ over the space L 2 (ΓX,πm), where πm is the Poisson measure on ΓX with intensity m—the volume measure on X. We construct a metric space Γ ∞ that is continuously embedded into ΓX. Under some conditions on the manifold X, we prove that Γ ∞ is a set of full πm measure and derive an explicit formula for the heat semigroup: (e−tHΓF)(γ) = ∫ Γ ∞ F(ξ)Pt,γ(dξ), where Pt,γ is a probability measure on Γ ∞ for all t> 0, γ ∈ Γ∞. The central results of the paper are two types of Feller properties
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.
The maximum Markovian selfadjoint extension of Dirichlet operators for interacting particle systems
 Forum Math
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Some Exceptional Configurations
, 2000
"... The Dirichlet form given by the intrinsic gradient on Poisson space is associated with a Markov process consisting of a countable family of interacting di#usions. By considering each di#usion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure value ..."
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The Dirichlet form given by the intrinsic gradient on Poisson space is associated with a Markov process consisting of a countable family of interacting di#usions. By considering each di#usion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued di#usion. The quasisure analysis of Dirichlet forms is used to find exceptional sets for this Markov process. We show that the process never hits certain unusual configurations, such as those with more than unit mass at some position, or those that violate the law of large numbers. Some of these results also hold for Gibbs measures with superstable interactions. AMS (1991) subject classification 60H07, 31C25, 60G57, 60G60 1 Introduction In recent work [1, 2, 3, 4, 10, 11, 12, 14, 17] the theory of Dirichlet forms has been used to construct and study Markov processes that take values in the space #X of locally finite configurations on a Riemannian manifold X. The configuration space is d...