The strong solution of the Monge-Ampère equation on Wiener space for general log-concave measures
The strong solution of the Monge-Ampère equation on Wiener space for general log-concave measures
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一般对数凹测度维纳空间上 Monge-Ampère 方程的强解
DOI：
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发表时间：
2007
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通讯作者：
A. Üstünel
中科院分区：
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作者：
A. Üstünel
本文证明了MongeKantorovitch测度在Wiener测度和具有H-log-凹密度的目标测度之间的迁移问题的唯一1-凸解，在[9]的意义下，Wiener测度也是MongeAmpère方程在无限维Fréchet空间框架下的强解. w.r.to进一步推广了利用Monge-Kantorovitch测度迁移将一个扩散测度变换为另一个扩散测度的映射的极分解结果，并阐明了这一概念与Monge-Ampère方程的Itô-解之间的关系.
In this work we prove that the unique 1-convex solution of the MongeKantorovitch measure transportation problem between the Wiener measure and a target measure which has an H-log-concave density, in the sense of [9], w.r.to the Wiener measure is also the strong solution of the MongeAmpère equation in the frame of infinite dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge-Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge-Ampère equation.