Some Applications of Brenier Map and Diffusion Semigroup to Gaussian-Type Functional Inequalities
Date Issued
2014
Date
2014
Author(s)
Chin, You-Lin
Abstract
In this paper, we follow with these two results of [7] and [12] documents, explaining some of its contents again. We prove the contents of some of its more than supplement of the theory, as much as possible to provide some prior knowledge and other related theories, as the link between theories. As a result, this paper presents a more complete knowledge of mathematics. First, we explore a map, discovered by Brenier, which is a convex gradient and gives the optimal mass transport (with cost function
c(x,y)=|x−y|^2) in R^n. This map can be used to derive some functional inequalities with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev inequalities, Talagrand’s transport inequalities and HWI inequality are recovered. Second, we investigate diffusion semigroups and using Bakry-Emery gamma2-criterium to obtain Poincar’e inequality and logarithmic Sobolev inequality. Finally, by using the previous map and diffusion semigroups to prove Gaussian correlation inequality under some conditions. To accomplish this work, we refer to the documents listed as Bibliography, there are mainly books, such as [27] and [17].
Subjects
最優質量運輸
不雷尼爾函數
奧恩斯坦-烏倫貝克半群
擴散半群
Bakry-Emery準則
Type
thesis
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