英文标题：
《On the K\\\"ahler Geometry of Certain Optimal Transport Problems》
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作者：
Gabriel Khan, Jun Zhang
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最新提交年份：
2019
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英文摘要：
Let $X$ and $Y$ be domains of $\\mathbb{R}^n$ equipped with respective probability measures $\\mu$ and $ \\nu$. We consider the problem of optimal transport from $\\mu$ to $\\nu$ with respect to a cost function $c: X \\times Y \\to \\mathbb{R}$. To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are strongly \\textit{relatively $c$-convex} with respect to each other and cost function has non-negative \\textit{MTW tensor}. For cost functions of the form $c(x,y)= \\Psi(x-y)$ for some convex function $\\Psi$, we find an associated K\\\"ahler manifold whose orthogonal anti-bisectional curvature is proportional to the MTW tensor. We also show that relative $c$-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\\\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to $\\mathbb{C}^2$. We also address a question in mathematical finance raised by Pal and Wong on the regularity of \\textit{pseudo-arbitrages}, or investment strategies which outperform the market.
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中文摘要：
设$X$和$Y$是$\\mathbb{R}^n$的域，分别具有$\\mu$和$\\nu$的概率测度。我们考虑了从$\\mu$到$\\nu$的最优传输问题，其代价函数为$\\c:X乘以Y到$\\mathbb{R}$。为了确保这个问题的解决是顺利的，有必要对域的结构和成本函数做出一些假设。特别地，马、特鲁丁格和王建立了正则性估计，当域之间是强凸的，且代价函数具有非负的MTW张量时。对于形式为$c（x，y）=\\ Psi（x-y）$的成本函数，对于某些凸函数$\\ Psi$，我们找到一个关联的K\\“ahler流形，其正交反等分曲率与MTW张量成比例。我们还表明，相对$c$-凸性在几何上对应于关于对偶仿射连接的测地凸性。综合起来，这些结果为优化传输提供了一个几何框架，这是对Kim和McCann的伪黎曼理论的补充。我们提供了几个ral本工程的应用。特别是，我们发现了一个具有非负正交二分曲率的完整K“ahler曲面，它不是Hermitian对称空间，也不是$\\mathbb{C}^2$的双全纯曲面。我们还解决了Pal和Wong在数学金融学中提出的一个问题，即伪套利的正则性，或优于市场的投资策略。
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分类信息：

一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Mathematics 数学
二级分类：Differential Geometry 微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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