Configuration Spaces over Singular Spaces -- II. Curvature

Abstract

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