Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures

TL;DR

This paper investigates the geometric and spectral properties of optimal transportation maps between log-concave measures, establishing conditions under which the induced metric-measure space has non-negative curvature and satisfies CD(K,N) conditions, leading to dimension-free estimates.

Contribution

It introduces a Riemannian metric based on the Hessian of the transportation potential and proves curvature and CD conditions for the resulting space, extending geometric analysis of optimal transport.

Findings

The space admits a non-negative Bakry–Émery tensor when potentials are convex.

Under certain conditions, the space is a CD(K,N) space, enabling geometric and probabilistic estimates.

Derived dimension-free bounds on the Hessian of the transportation map and diameter estimates for the metric space.

Abstract

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--{\'E}mery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.