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Analytic aspects of optimal transportation

最佳运输的分析方面

基本信息

批准号：
316972354
负责人：
Professor Dr. Michael Röckner
金额：
--
依托单位：
Arbeitsgruppe Stochastische Analysis
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/316972354?language=en
关键词：
Analytic aspects optimal transportation

项目摘要

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.

该项目的主要目标是调查分析方面的新问题在理论的最佳运输有关的各种限制最佳映射或最佳计划。所考虑的问题有显着的重要性，一些领域的数学和应用。预计将取得以下成果。带附加限制的运输问题的Kantorovich对偶性及其与遍历理论的关系。考虑鞅输运问题、一般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.构造一个非交换的Monge-Kantorovich理论.能量测度空间及相关输运问题的研究.研究无限维空间上最优运输理论的动力学问题。最优计划的转换。具有最优运输方案的空间上非线性泛函分布的绝对连续性条件.利用康托洛维奇度量在概率测度空间中优化控制梯度流。6. Kähler-Einstein方程的真实的版本的研究。其中一个可能的应用是获得凸体等周常数的最佳可能渐近界。计划研究Kähler-Einstein方程的先验估计和各种几何特征，特别是解的三阶导数的估计。研究Wiener空间上的无限维真实的Kähler-Einstein方程，其中镜像测度与Wiener测度一致。计划证明无限维情况下Kähler-Einstein方程解的存在性，其中最佳传输由某一测量的对数梯度给出。研究由最优输运势诱导的具有测度和Hessian度量的流形。主要预期应用是等周常数和索博列夫常数的新估计、康托洛维奇距离的界限（传输不等式）。9.具有多个（多于两个）边值的运输问题。我们计划获得精确的描述的解决方案的运输问题的一维marginalsand的成本函数是最小的仿射函数。我们还计划将第1项中提到的一些结果推广到更多的边缘人群。10.用最优输运法获得凸体的新几何不等式。