We define a de Rham complex over a product manifold (infinite product of compact manifolds), and Dirichlet operators on differential forms, associated with differentiable measures (in particular, Gibbs measures), which generalize the notions of Bochner and de Rham Laplacians. We give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics.

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given. 2000

The distribution μ cl of a Poisson cluster process in X = R d (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X = n X n , with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure μ cl is quasi-invariant with respect to the group of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for μ cl . The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.

Gibbs states of a spin system with the single-spin space S = R m and unbounded pair interactions is studied. The spins are attached to the points of a realization γ of a random point process in R n . Under certain conditions on the model parameters we prove that, for almost all γ, the set G(S γ ) of all Gibbs states is nonempty and its elements have support properties, explicitly described in the paper. We also show the existence of measurable selections γ → νγ ∈ G(S γ ) (random Gibbs measures) and derive the corresponding averaged moment estimates.

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over R d , d ≥ 2. Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the intensity of the underlying point process and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.