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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
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dc.contributor.advisor | 王藹農(Ai-Nung Wang) | |
dc.contributor.author | You-Lin Chin | en |
dc.contributor.author | 金宥霖 | zh_TW |
dc.date.accessioned | 2021-06-16T08:06:56Z | - |
dc.date.available | 2014-07-17 | |
dc.date.copyright | 2014-07-16 | |
dc.date.issued | 2014 | |
dc.date.submitted | 2014-06-17 | |
dc.identifier.citation | [1] C’ecile An’e, S’ebastien Blach`ere, Djalil Chafai‥, Pierre Foug`eres, Ivan Gentil, Florent Malrieu, Cyril Roberto, and Gr’egory Scheffer. Sur les in’egalit’es de
Sobolev logarithmiques, volume 10 of Panoramas et Synth`eses [Panoramas and Syntheses]. Soci’et’e Math’ematique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux. [2] Dominique Bakry. Functional inequalities for markov semigroups. Probability measures on groups: recent directions and trends, pages 91–147, 2006. [3] Dominique Bakry and Michel ’Emery. Diffusions hypercontractives. In S’eminaire de Probabilit’es XIX 1983/84, pages 177–206. Springer, 1985. [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. [8] G. Da Prato and J. Zabczyk. Ergodicity for infinite-dimensional systems, volume 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1996. [9] Guido De Philippis and Alessio Figalli. The monge- ampere equation and its link to optimal transportation. arXiv preprint arXiv:1310.6167, 2013. [10] Stewart N Ethier and Thomas G Kurtz. Markov processes: characterization and convergence, volume 282. John Wiley & Sons, 2009. [11] Lawrence C. Evans and Ronald F. Gariepy. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [12] Ivan Gentil. Logarithmic sobolev inequality for diffusion semigroups. arXiv preprint arXiv:1009.3421, 2010. [13] Leonard Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97(4):1061–1083, 1975. [14] A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In S’eminaire de Probabilit’es, XXXVI, volume 1801 of Lecture Notes in Math., pages 1–134. Springer, Berlin, 2003. [15] Gilles Harg’e. A particular case of correlation inequality for the Gaussian measure. Ann. Probab., 27(4):1939–1951, 1999. [16] Alexander V Kolesnikov. Mass transportation and contractions. arXiv preprint arXiv:1103.1479, 2011. [17] Luca Lorenzi and Marcello Bertoldi. Analytical methods for Markov semigroups, volume 283 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007. [18] Robert J McCann. Existence and uniqueness of monotone measure-preserving maps. Duke Mathematical Journal, 80(2):309–324, 1995. [19] Robert J McCann. A convexity principle for interacting gases. advances in mathematics, 128(1):153–179, 1997. [20] Gaspard Monge. M’emoire sur la th’eorie des d’eblais et des remblais. Histoire de l’Acad’emie royale des sciences de Paris, pages 666–704, 1784. [21] Felix Otto and C’edric Villani. Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. Journal of Functional Analysis, 173(2):361–400, 2000. [22] A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. [23] Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. [24] G Schechtman, Th Schlumprecht, and J Zinn. On the gaussian measure of the intersection of symmetric, convex sets. preprint. [25] M. Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal., 6(3):587–600, 1996. [26] Adrian D. Tudorascu. Optimal mass transportation methods for gradient flows in the weak topology. ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–Carnegie Mellon University. [27] C’edric Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/58149 | - |
dc.description.abstract | 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]。 | zh_TW |
dc.description.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]. | en |
dc.description.provenance | Made available in DSpace on 2021-06-16T08:06:56Z (GMT). No. of bitstreams: 1 ntu-103-R97221002-1.pdf: 858319 bytes, checksum: 5d78288619d87b97dc05a9f2643002b3 (MD5) Previous issue date: 2014 | en |
dc.description.tableofcontents | Table of Contents iv
1 Introduction 1 2 Brenier Map 4 2.1 Preliminaries of Brenier map . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Applications of Brenier map to inequalities involving entropy . . . . . 12 3 Diffusion Semigroup 27 3.1 The Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . . . . . . 27 3.2 Applications of semigroup to Poincar’e and Logarithmic Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4 Applications of Brenier map and Semigroup to Gaussian correlation inequality 48 4.1 Ca.arelli’s regularity theorem and contraction theorem . . . . . . . . 48 4.2 Gaussian correlation inequality . . . . . . . . . . . . . . . . . . . . . 49 5 Appendix 52 5.1 Second derivatives almost everywhere for convex functions . . . . . . 52 5.2 Markov semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Bibliography 60 | |
dc.language.iso | zh-TW | |
dc.title | 不雷尼爾函數跟擴散半群應用於高斯型泛函不等式 | zh_TW |
dc.title | Some Applications of Brenier Map and Diffusion Semigroup to Gaussian-Type Functional Inequalities | en |
dc.type | Thesis | |
dc.date.schoolyear | 102-2 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 夏俊雄(Chun-Hsiung Hsia),陳界山(Jein-Shan Chen) | |
dc.subject.keyword | 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, | zh_TW |
dc.subject.keyword | Optimal mass transport,Brenier map,Diffusion Semigroup,Ornstein-Uhlenbeck semigroup,Bakry-Emery criterium, | en |
dc.relation.page | 62 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2014-06-17 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 數學研究所 | zh_TW |
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