Configuration Spaces over Singular Spaces
A talk in the Bielefeld Stochastic Afternoon series by
Kohei Suzuki
| Abstract: | Please contact stochana@math.uni-bielefeld.de for Meeting-ID and Password.
The configuration space Y(X) over the base space X is the space of all locally finite point measures on X, which has been studied in various areas of mathematics: interacting particle systems in statistical physics; infinite-dimensional metric measure geometry; representation theory of diffeomorphism groups etc. In this talk, we focus on analytic and geometric structure on Y(X). We first present an analytic structure on Υ(X), constructing a strongly local Dirichlet form for a large class of invariant measures µ. We then investigate the geometric structure of the extended metric measure space Υ(X) endowed with the L^2-transportation extended distance d_2 and with the measure µ. By establishing various Rademacher- and Sobolev-to-Lipschitz-type properties, we finally provide a complete identification of the analytic and the geometric structure — the canonical differential structure induced on Υ(X) by X and µ. The class of base spaces we discuss includes RCD spaces, locally doubling metric measure spaces satisfying a local Poincaré inequality, and sub-Riemannian manifolds; as for µ our results include Campbell measures and quasi-Gibbs measures, in particular: Poisson measures, canonical Gibbs measures, as well as some determinantal/permanental point processes (sineβ, Airyβ, Besselα,β, Ginibre). A number of applications to interacting particle systems and infinite-dimensional metric measure geometry are also discussed. Our approach does not rely on any relation between µ and a possible smooth structure on X. In particular, we assume no quasi-invariance property of µ w.r.t. actions of any group of transformations of X. As a consequence, several of our results are new even in the case of standard Euclidean spaces. |