Analytic aspects of optimal transportation

• Applicant: Professor Dr. Michael Röckner
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 316972354

The chief goal of the project is investigation of analytic aspects of new problems in thetheory of optimal transportation related to diverse restrictions on optimal mappings oron optimal plans. The considered problem have significant importance for a number ofareas of mathematics and applications. It is envisaged to obtain the following results.1. A study of the Kantorovich duality for transport problems with additional restrictionsand its relations to ergodic theory. Consideration of martingale transport problems,general linear restrictions, and some other types of restrictions. Investigation of the structureof transport plans generated by ergodic decompositions.2. Constructing a noncommutative Monge-Kantorovich theory.3. A study of energy measure spaces and related transportation problems.4. Investigation of dynamical problems of the theory of optimal transportation oninfinite-dimensional spaces. Transformations of optimal plans. Conditions for the absolutecontinuity of the distributions of nonlinear functionals on spaces with optimaltransportation plans.5. Optimal control of gradient flows in the space of probability measures with theKantorovich metric.6. Investigation of the real version of the Kähler-Einstein equation. One of possibleapplications is obtaining best possible asymptotic bounds for isoperimetric constants ofconvex bodies. It is planned to study a priori estimates and various geometric characteristicsof the Kähler-Einstein equation, in particular, estimates for the third order derivativesof solutions.7. Investigation of the infinite-dimensional real Kähler-Einstein equation on the Wienerspace, where the image measure coincides with the Wiener measure. It is planned to provethe existence of a solution to the Kähler-Einstein equation in the infinite-dimensional case,where the optimal transport is given by the logarithmic gradient of a certain measure.8. Investigation of manifolds equipped with measures and Hessian metrics inducedby potentials of optimal transports. The main expected applications are new estimatesof isoperimetric constants and Sobolev constants, bounds for the Kantorovich distance(transport inequalities).9. The transport problem with many (more than two) marginals. We plan to obtain aprecise description of solutions to the transport problem with one-dimensional marginalsand the cost function that is the minimum of affine functions. We also plan to generalizesome results mentioned in item 1 to a larger number of marginals.10. Obtaining new geometric inequalities for convex bodies by means of optimal transportation.

• DFG Programme: Research Grants