Inserted: 16 dec 2015
Last Updated: 16 dec 2015
Journal: Calculus of Variations and Partial Differential Equations
Aim of this paper is to show that it makes sense to write the continuity equation on a metric measure space $(X,d,m)$ and that absolutely continuous curves $(\mu_t)$ w.r.t. the distance $W_2$ can be completely characterized as solutions of the continuity equation itself, provided we impose the condition $\mu_t\leq Cm$ for every $t$ and some $C>0$.
Keywords: Optimal transport, continuity equation, metric measure spaces, absolutely continuous curve