Configuration Spaces over Singular Spaces -- II. Curvature

Abstract

This is the second paper of a series on configuration spaces ${\rm Y}$ over singular spaces $X$. Here, we focus on geometric aspects of the extended metric measure space $({\rm Y},{\mathsf{d}}_{{\rm Y}},\mu )$ equipped with the $L^2$-transportation distance ${\mathsf{d}}_{{\rm Y}}$, and a mixed Poisson measure $\mu$. Firstly, we establish the essential self-adjointness and the $L^p$-uniqueness for the Laplacian on ${\rm Y}$ lifted from $X$. Secondly, we prove the equivalence of Bakry-Émery curvature bounds on $X$ and on ${\rm Y}$, without any metric assumption on $X$. We further prove the Evolution Variation Inequality on ${\rm Y}$, and introduce the notion of synthetic Ricci-curvature lower bounds for the extended metric measure space ${\rm Y}$. As an application, we prove the Sobolev-to-Lipschitz property on ${\rm Y}$ over singular spaces $X$, originally conjectured in the case when $X$ is a manifold by M. Röckner and A. Schield. As a further application, we prove the $L^{\infty }$-to-${\mathsf{d}}_{{\rm Y}}$-Lipschitz regularization of the heat semigroup on ${\rm Y}$ and gives a new characterization of the ergodicity of the corresponding particle systems in terms of optimal transport.

50 pages