*Published Paper*

**Inserted:** 16 dec 2015

**Last Updated:** 16 dec 2015

**Journal:** Calculus of Variations and Partial Differential Equations

**Volume:** 53

**Pages:** 149-177

**Year:** 2015

**Abstract:**

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

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