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

We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker–Planck equations, when the drift is a 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. This technique 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 and Otto (1998), to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.