Abstract
The Lagrangian description of absolutely continuous 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 to 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 to a given velocity, i.e. they provide uniqueness for the solution of the continuity equation within a certain class of curves.
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