Isoperimetric minimizing movements and AC curves in spaces of measures

Abstract

We define a complete metric structure on the family ${\text{PL}}_q^p({\mathbb{R}}^n)$ of probability measures with densities in $L^p({\mathbb{R}}^n)$ and finite $q$-moments. We establish the existence of generalized minimizing movements for the isoperimetric ratio and characterize absolutely continuous curves in this space through weak solutions of the continuity equation with velocity fields satisfying a first-order integral condition. We also characterize absolutely continuous curves in the $\infty$-Wasserstein space and prove a Benamou--Brenier formula for $W_{\infty }$.

Key words: Wasserstein spaces, isoperimetric problem, minimizing movement schemes