On the K\"ahler Geometry of Certain Optimal Transport Problems

Gabriel Khan and Jun Zhang

Papers from arXiv.org

Abstract: 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.

Date: 2018-11, Revised 2019-08
References: View references in EconPapers View complete reference list from CitEc
Citations:

Published in Pure Appl. Analysis 2 (2020) 397-426

Downloads: (external link)
http://arxiv.org/pdf/1812.00032 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1812.00032

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().