The distribution µ of a Poisson cluster process in X = R d (with n-point 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 quasi-invariant 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.

The space ΓX of all locally finite configurations in a Riemannian manifold X of infinite volume is considered. The deRham complex of square-integrable 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 2-cohomology of the underlying manifold X.

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...

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

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The distribution µ of a Gibbs cluster point process in X = R d (with n-point 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 quasi-invariant with respect to the group Diff0(X) of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for µ. The corresponding equilibrium stochastic dynamics is then constructed by using the method of Dirichlet forms.