不雷尼爾函數跟擴散半群應用於高斯型泛函不等式

Some Applications of Brenier Map and Diffusion Semigroup to Gaussian-Type Functional Inequalities

金宥霖(You-Lin Chin)

指導教授 ： 王藹農

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.01589

摘要

在這篇論文中,我們依循著這兩篇文獻[7]和[12]的成果,將其部分內容闡述一遍,並將其某些理論之證明內容多作補充,也盡可能提供部分先備知識和其他相關理論,以作為理論之間的連結,讓這篇論文呈現出一套較完備的數學知識。首先,我們探索一個函數,被不雷尼爾發現,這個函數是一個凸函數的梯度並為「質量運輸問題」(當花費函數c(x,y)=|x−y|^2時)在n維實數空間的極值解。藉由這個函數的質量運輸性直接的推導出幾個泛函不等式,得出的結果有「對數索伯列夫不等式」、「Talagrand 運輸不等式」、「HWI不等式」。然後,我們探討擴散半群並且藉由 「Bakry-Emery gamma2-準則」推導出「龐加萊不等式」跟「對數索伯列夫不等式」。 最後,利用先前介紹的函數和擴散半群證明了在某些條件下的「高斯相關不等式」。為了完成這次工作,我們參考的文獻如Bibliography所列,其中主要書籍
有[27]和[17]。

關鍵字

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

並列摘要

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].

並列關鍵字

Optimal mass transport ； Brenier map ； Diffusion Semigroup ； Ornstein-Uhlenbeck semigroup ； Bakry-Emery criterium

參考文獻

[2] Dominique Bakry. Functional inequalities for markov semigroups. Probability measures on groups: recent directions and trends, pages 91–147, 2006.

[4] Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on pure and applied mathematics, 44(4):375–417, 1991.

[5] Luis A. Caffarelli. The regularity of mappings with a convex potential. J. Amer. Math. Soc., 5(1):99–104, 1992.

[6] Luis A. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys., 214(3):547–563, 2000.

[7] Dario Cordero-Erausquin. Some applications of mass transport to Gaussiantype inequalities. Arch. Ration. Mech. Anal., 161(3):257–269, 2002.