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

TL;DR

This paper develops a canonical differential structure on configuration spaces over singular base spaces, linking analytic and geometric structures via Dirichlet forms and metric measure geometry, with broad applications to particle systems and infinite-dimensional spaces.

Contribution

It constructs a canonical differential structure on configuration spaces over singular spaces, unifying analytic and geometric frameworks through Dirichlet forms and metric measure theory.

Findings

Constructed a strongly local Dirichlet form on configuration spaces.

Established the equivalence of analytic and geometric structures on the space.

Proved universality of the $L^2$-transportation distance for diffusion asymptotics.

Abstract

We construct a canonical differential structure on the configuration space $\Upsilon$ over a singular base space $X$ and with a general invariant measure $\mu$ on $\Upsilon$. We present an analytic structure on $\Upsilon$, constructing a strongly local Dirichlet form $\mathcal E$ on $L^2(\Upsilon, \mu)$ for $\mu$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space $\Upsilon$ endowed with the $L^2$-transportation extended distance $\mathsf{d}_{\Upsilon}$ 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 $\Upsilon$ by $X$ and $\mu$ -- showing that $\mathcal E$ coincides with the Cheeger energy of $(\Upsilon,\mathsf{d}_{\Upsilon},\mu)$ and that the intrinsic distance of $\mathcal E$ coincides with $\mathsf{d}_{\Upsilon}$. 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 $\mathsf{d}_{\Upsilon}$ for the Varadhan short-time asymptotics for diffusions on $\Upsilon$, regardless of the choice of $\mu$. Many of our results are new even in the case of configuration spaces over Euclidean spaces.