Sobolev estimates for optimal transport maps on Gaussian spaces |
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Authors: | Shizan Fang Vincent Nolot |
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Institution: | I.M.B., BP 47870, Université de Bourgogne, Dijon, France |
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Abstract: | In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained. |
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Keywords: | Optimal transportation Sobolev estimates Gaussian measures Monge&ndash Ampè re equations Wiener space |
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