Calculus of Variations and Geometric Measure Theory

S. Lisini

Characterization of absolutely continuous curves in Wasserstein spaces

created by lisini on 21 Feb 2008


Published Paper

Inserted: 21 feb 2008

Journal: Calc. Var.
Volume: 28
Pages: 85-120
Year: 2007


Let $X$ be a separable, complete metric space and $P_p(X)$ be the space of Borel probability measures with finite moment of order $p>1$, metrized by the Wasserstein distance. In this paper we prove that every absolutely continuous curve with finite $p$-energy in the space $P_p(X)$ can be represented by a Borel probability measure on $C([0,T];X)$ concentrated on the set of absolutely continuous curves with finite $p$-energy in $X$. Moreover this measure satisfies a suitable property of minimality which entails an important relation on the energy of the curves. We apply this result to the geodesics of $P_p(X)$ and to the continuity equation in Banach spaces.