Curvature bounds for configuration spaces

TL;DR

This paper demonstrates that configuration spaces over manifolds inherit key curvature properties from the base manifold, including Ricci bounds, inequalities, and flow behaviors, linking geometric analysis with probabilistic processes.

Contribution

It establishes that curvature bounds and related geometric properties of a manifold extend to its configuration space, connecting differential geometry with stochastic analysis.

Findings

Configuration space inherits Ricci curvature bounds from the manifold.

Heat flow on configuration space is identified as the gradient flow of entropy.

Configuration space satisfies Bochner inequality and Wasserstein contraction.

Abstract

We show that the configuration space over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies for the configuration space a lower Ricci curvature bound in the sense of Lott-Sturm-Villani, the Bochner inequality, gradient estimates and Wasserstein contraction. Moreover, we show that the heat flow on the configuration space, or the infinite independent particle process, can be identified as the gradient flow of the entropy.