Flows of measures induced by measurable vector fields

Eugene Stepanov

(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.