preprint

Inserted: 2 jul 2025

Year: 2025

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.