Regularized Moment Measures

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 $(\mathrm{\nabla }u{)}_{\mathrm{♯}}{e}^{-u}=\mu$. 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 $(\mathrm{\nabla }u{)}_{\mathrm{♯}}{e}^{-u-\frac{\alpha }{2}\mathrm{\parallel }x{\mathrm{\parallel }}^2}=\mu$. Exploiting the regularization term, we study the stability of the minimizers.