Analytic aspects of optimal transportation

该项目的主要目标是研究最优运输理论中与对最优映射或最优计划的各种限制有关的新问题的分析方面。所考虑的问题对数学和应用的许多领域具有重要意义。预计将取得以下成果1。具有附加限制的运输问题的Kantorovich对偶性及其与遍历理论的关系。讨论了鞅运输问题、一般的线性约束和一些其他类型的约束。研究遍历分解生成的运输计划的结构。构建非对易的Monge-Kantorovich理论。研究能源计量空间及相关交通问题。无限维空间上最优运输理论的动力学问题研究。最优方案的变换。具有最优运输方案的空间上非线性泛函分布绝对连续的条件。基于Kantorovich度量的概率度量空间中梯度流的最优控制。卡勒-爱因斯坦方程的真实版本的研究。其中一个可能的应用是获得凸体等周常数的最佳可能的渐近界。计划研究Kähler-Einstein方程的先验估计和各种几何性质，特别是解的三阶导数的估计。研究了维纳空间上无限维实Kähler-Einstein方程的像测度与维纳测度重合的情形。证明了Kähler-Einstein方程在无限维情形下解的存在性，其中最优输运是由某一测度的对数梯度给出的。具有度量的流形和由最优传输势诱导的黑森度量的研究。主要的预期应用是等周常数和索博列夫常数的新估计，即Kantorovich距离的界(传输不等式)。有许多(两个以上)边缘的运输问题。我们计划得到一维边缘运输问题的解的精确描述，以及作为仿射函数的最小值的代价函数。我们还计划将项目1中提到的可喜结果推广到更多的边缘。利用最优输运方法得到凸体的新几何不等式。

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.