Traces of semigroups associated with interacting particle systems

Abstract

Witten Laplacians associated with interacting particle systems on infinite coverings of compact manifolds are considered. The probabilistic representations of the corresponding heat kernels are given. Von Neumann traces of the corresponding semigroups are computed.

L2 dimensions of spaces of braid-invariant harmonic forms

“…The right-hand side of formula (45) can be understood as a regularized index of the Dirac operator D + D * , where D := m N d m N , see [11] and e.g. [30] for the discussion of von Neumann supertraces in geometry and topology of Riemannian manifolds and their relation to L 2 -index theorems, and [1], [2], [3] for the extension of these notions to the framework of infinite configuration spaces.…”

Gibbs cluster measures on configuration spaces

“…Albeverio, Kondratiev and Röckner [2,3] have proposed an approach to configuration spaces Γ X as infinite-dimensional manifolds, based on the choice of a suitable probability measure μ on Γ X which is quasi-invariant with respect to Diff 0 (X), the group of compactly supported diffeomorphisms of X. Providing that the measure μ can be shown to satisfy an integration-by-parts formula, one can construct, using the theory of Dirichlet forms, an associated equilibrium dynamics (stochastic process) on Γ X such that μ is its invariant measure [2,3,31] (see [1,4,11,15,22,23,34,39] and references therein for further discussion of various theoretical aspects and applications).…”

Cluster point processes on manifolds

“…Such a programme has been implemented for the Poisson and Gibbs measures on Γ X in the case where X = R d (see [3,4,5,2,1] and further references therein). The present paper is a step towards realisation of this programme for the important class of (in general, non-Gibbsian) measures on Γ X emerging as distributions of cluster point processes in X.…”