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On absolutely continuous curves of probabilities on the line

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  • In recent collaborative work we studied existence and uniqueness of a Lagrangian description for absolutely continuous curves in spaces of Borel probabilities on the real line with finite moments of given order. Of course, a measurable velocity driving the evolution in Eulerian coordinates is necessary to define the Eulerian and Lagrangian descriptions of fluid flow; here we prove that in this case it is also sufficient for a Lagrangian description. More precisely, we argue that the existence of the integrable velocity along an absolutely continuous curve in the set of Borel probabilities on the line is enough to produce a canonical Lagrangian description for the curve; this is given by the family of optimal maps between the uniform distribution on the unit interval and the measures on the curve. Moreover, we identify a necessary and sufficient condition on said family of optimal maps which ensures that the measurable velocity along the curve exists.

    Mathematics Subject Classification: 34A12, 34A34, 35A02, 35A24, 35F20, 35Q35, 60E05, 76A02.


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