Absolutely continuous curves in spaces of compactly supported densities

Abstract

We give a constructive proof for existence of absolutely continuous curves connecting each pair $\mu ,\nu \in {\mathrm{PL}}_{\infty }^p({\mathbb{R}}^n)$, for every $1\le p\le \infty$, where $({\mathrm{PL}}_{\infty }^p({\mathbb{R}}^n),{\mathfrak{d}}_{\infty }^p)$ is the complete metric space of absolutely continuous measures with density in $L^p({\mathbb{R}}^n)$ and bounded support introduced in [1].

arXiv commentKey words: Wasserstein spaces, absolute continuity, spaces of measures