Heat flow, log-concavity, and Lipschitz transport maps

TL;DR

This paper establishes bounds on the log-Hessian for heat equation solutions with log-Lipschitz initial data, leading to new Lipschitz estimates for transport maps and insights into diffusion models and inequalities.

Contribution

It provides explicit lower bounds for the log-Hessian in heat flow with log-Lipschitz initial data, advancing understanding of transport maps and inequalities.

Findings

Uniform lower bounds for log-Hessian under specific initial conditions

New Lipschitz estimates for transport maps from Gaussian to log-Lipschitz measures

Fast decay of tails alone does not ensure uniform log-Hessian bounds

Abstract

In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. Further connections are discussed with score-based diffusion models and improved Gaussian logarithmic Sobolev inequalities. Finally, we show that assuming only fast decay of the tails of the initial datum does not suffice to guarantee uniform log-Hessian upper bounds.