Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts

L. Natile, M.A. Peletier, G. Savaré

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Abstract

We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in Rd, when the drift is a monotone (or lambda-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solutions without requiring stronger regularity and it extends the Wasserstein theory of Fokker-Planck equations with gradient drift terms started by Jordan-Kinderlehrer-Otto [14] to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.
Original languageEnglish
PublisherarXiv.org
Number of pages18
Publication statusPublished - 2010

Publication series

NamearXiv.org [math.AP]
Volume1002.0088

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