Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry

Abstract

We construct a canonical differential structure on the configuration space ${\rm Y}$ over a singular base space $X$ and with a general invariant measure $\mu$ on ${\rm Y}$. We present an analytic structure on ${\rm Y}$, constructing a strongly local Dirichlet form $\mathcal{E}$ on $L^2({\rm Y},\mu )$ for $\mu$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space ${\rm Y}$ endowed with the $L^2$-transportation extended distance ${\mathsf{d}}_{{\rm Y}}$ and with the measure $\mu$. By establishing Rademacher- and Sobolev-to-Lipschitz-type properties for $\mathcal{E}$, we finally provide a complete identification of the analytic and the geometric structure -- the canonical differential structure induced on ${\rm Y}$ by $X$ and $\mu$ -- showing that $\mathcal{E}$ coincides with the Cheeger energy of $({\rm Y},{\mathsf{d}}_{{\rm Y}},\mu )$ and that the intrinsic distance of $\mathcal{E}$ coincides with ${\mathsf{d}}_{{\rm Y}}$. The class of base spaces to which our results apply includes sub-Riemannian manifolds, RCD spaces, and path/loop spaces over Riemannian manifolds; as for $\mu$ our results include quasi-Gibbs measures, in particular: Poisson measures, canonical Gibbs measures, as well as some determinantal/permanental point processes (sine${}_{\beta }$, Airy${}_{\beta }$, Bessel${}_{\alpha ,\beta }$, Ginibre). A number of applications to interacting particle systems and infinite-dimensional metric measure geometry are also discussed. In particular, we prove the universality of the $L^2$-transportation distance ${\mathsf{d}}_{{\rm Y}}$ for the Varadhan short-time asymptotics for diffusions on ${\rm Y}$, regardless of the choice of $\mu$. Many of our results are new even in the case of configuration spaces over Euclidean spaces.

85 pages, 3 diagrams, 3 tables (minor modifications, added table 3)