On the Lagrangian Description of Absolutely Continuous Curves in the Wasserstein Space on the Line; Well-Posedness for the Continuity Equation

The Lagrangian description of absolutely continnous curves of probability measures on the real line is analyzed. Whereas each such curve admits a Lagrangian description as a well-defined flow of its velocity field, further conditions on the curve and/or its velocity are necessary for uniqueness. We identify two seemingly unrelated such conditions that ensure that the only flow map associated with the curve consists of a time-independent rearrangement of the generalized inverses of the cumulative distribution functions of the measures on the curve. At the same time, our methods of proof yield uniqueness within a certain class for the curve associated with a given velocity; that is, they provide uniqueness for the solution of the continuity equation within a certain class of curves.