Contraction of general transportation costs along solutions to Fokker–Planck equations with monotone drifts
✍ Scribed by Luca Natile; Mark A. Peletier; Giuseppe Savaré
- Publisher
- Elsevier Science
- Year
- 2011
- Tongue
- English
- Weight
- 208 KB
- Volume
- 95
- Category
- Article
- ISSN
- 0021-7824
No coin nor oath required. For personal study only.
✦ Synopsis
We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in R d , when the drift is a monotone (or λ-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 and Otto (1998) [14], to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.