Variational problems involving unequal dimensional optimal transport
Variational problems involving unequal dimensional optimal transport
复制标题
涉及不等维最优输运的变分问题
DOI：
10.1016/j.matpur.2020.05.004
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Brendan Pass
中科院分区：
文献类型：
--
作者：
Luca Nenna;Brendan Pass
本文主要研究不等维空间之间的最优输运的概率测度集上的变分问题。特别是，我们研究了一个功能的总和，反映了一个固定的和一个自由的边际之间的成本（不平等的维度）最优运输，另一个功能的自由边际（各种形式）。激励应用包括古诺-纳什均衡，其中策略空间比代理类型的空间维度更低。对于各种不同形式的上述长期，我们表明，嵌套条件，这是已知的最优运输问题产生更好的易处理性，持有任何极小。根据确切的形式的功能，我们利用这一点找到局部微分方程的特征解决方案，证明收敛的迭代方案来计算的解决方案，并证明正则性结果。
This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal (of various forms). Motivating applications include Cournot-Nash equilibria where the strategy space is lower dimensional than the space of agent types. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. Depending on the exact form of the functional, we exploit this to find local differential equations characterizing solutions, prove convergence of an iterative scheme to compute the solution, and prove regularity results.