Regularized Moment Measures

TL;DR

This paper introduces a regularized variational approach to the moment measure problem, linking solutions to a modified equation and analyzing the stability of minimizers through added convex regularization.

Contribution

It extends Santambrogio's optimal transport framework by incorporating a strongly convex regularization, enabling stability analysis of solutions to the modified moment measure equation.

Findings

Regularization ensures stability of solutions.

Modified problem links to a new moment measure equation.

Provides insights into the stability of minimizers.

Abstract

In the work "Dealing with moment measures via entropy and optimal transport", Santambrogio provided an optimal transport approach to study existence of solutions for the moment measure equation, that is: given $\mu$, find $u$ such that $ (\nabla u)_{\sharp}e^{-u}=\mu$. In particular he proves that $u$ satisfies the previous equation if and only if $e^{-u}$ is the minimizer of an entropy and a transport cost. Here we study a modified minimization problem, in which we add a strongly convex regularization depending on a positive $\alpha$ and we link its solutions to a modified moment measure equation $(\nabla u)_{\sharp}e^{-u-\frac{\alpha}{2} \|x\|^2}= \mu$. Exploiting the regularization term, we study the stability of the minimizers.