# Flows of measures induced by measurable vector ﬁelds

### (St. Petersburg)

**Event:** ERC Workshop on Optimal Transportation and Applications

**Date:** Oct 28, 2014,
**time:** 14:30

**Abstract.** A smooth vector ﬁeld (say, over a mainfold) may be deﬁned either as a linear operator on the algebra of smooth functions satisfying Leibniz rule, or, equivalently, as a smooth ﬁeld of directions of curves (i.e. ”vectors”) at every point. The ﬁrst notion easily generalizes to what is known as measurable vector ﬁelds introduced by N. Weaver. These vector ﬁleds can in fact be identiﬁed with one-dimensional metric currents of Ambrosio and Kirchheim. We show that also the second identiﬁcation (with rectiﬁable curves in 3 place of smooth ones) is valid for a large class of measurable vector ﬁleds (but not all of them) and study the analogues of integral curves and ODE’s produced by such vector ﬁelds as well as the ﬂows of measures generated by them.