A note on Monge-Kantorovich problem

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• 作者：Pengbin Feng ; Xuhui Peng
• 关键词：Monge&ndash ; Kantorovich problem ; Optimal transportation ; Partial differential equations
• 刊名：Statistics and Probability Letters
• 出版年：January, 2014
• 年：2014
• 卷：84
• 期：Complete
• 页码：204-211
• 全文大小：385 K

文摘

Shen and Zheng (2010) and Xu and Yan (2013) considered the Monge-Kantorovich problem in the plane and proved that the optimal coupling for the problem has a form (, ), and then they assumed has a density and gave the equation which should satisfy. In this article, we prove that naturally has a density under more weak conditions. We again prove a similar result in dimension 3 and give an exact form (, ) depending on a certain convex function.