THÃSE KolÃ©hÃ¨ Abdoulaye COULIBALY-PASQUIER

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132 BIBLIOGRAPHY[ES95] Lawrence C. Evans and Joel Spruck. Motion of level sets by meancurvature. IV. J. Geom. Anal., 5(1):77–114, 1995.[ÉS99a][ES99b]M. Émery and W. Schachermayer. Brownian filtrations are not stableunder equivalent time-changes. In Séminaire de Probabilités, XXXIII,volume 1709 of Lecture Notes in Math., pages 267–276. Springer, Berlin,1999.L. C. Evans and J. Spruck. Motion of level sets by mean curvature. I[ MR1100206 (92h:35097)]. In Fundamental contributions to the continuumtheory of evolving phase interfaces in solids, pages 328–374.Springer, Berlin, 1999.[ESS92] L. C. Evans, H. M. Soner, and P. E. Souganidis. Phase transitionsand generalized motion by mean curvature. Comm. Pure Appl. Math.,45(9):1097–1123, 1992.[EY93][FL][GHL04][GZ07]K. D. Elworthy and M. Yor. Conditional expectations for derivatives ofcertain stochastic flows. In Séminaire de Probabilités, XXVII, volume1557 of Lecture Notes in Math., pages 159–172. Springer, Berlin, 1993.S. Fang and D. Luo. Quasi-invariant flows associated to tangent processeson the wiener space. J. Funct. Analysis (2007), no.2, 647–674.Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine. Riemanniangeometry. Universitext. Springer-Verlag, Berlin, third edition, 2004.Fuzhou Gong and Jingxiao Zhang. Flows associated to adapted vectorfields on the Wiener space. J. Funct. Anal., 253(2):647–674, 2007.[Ham82] Richard S. Hamilton. Three-manifolds with positive Ricci curvature. J.Differential Geom., 17(2):255–306, 1982.[Hsu95][Hsu02]Elton P. Hsu. Quasi-invariance of the Wiener measure on the path spaceover a compact Riemannian manifold. J. Funct. Anal., 134(2):417–450,1995.Elton P. Hsu. Stochastic analysis on manifolds, volume 38 of GraduateStudies in Mathematics. American Mathematical Society, Providence,RI, 2002.[Hui84] Gerhard Huisken. Flow by mean curvature of convex surfaces intospheres. J. Differential Geom., 20(1):237–266, 1984.[IW89]Nobuyuki Ikeda and Shinzo Watanabe. Stochastic differential equationsand diffusion processes, volume 24 of North-Holland Mathematical Library.North-Holland Publishing Co., Amsterdam, second edition, 1989.132
BIBLIOGRAPHY 133[Jos84][Jos05][JS03][Ken86][KN96][Kun90]Jürgen Jost. Harmonic mappings between Riemannian manifolds, volume4 of Proceedings of the Centre for Mathematical Analysis, AustralianNational University. Australian National University Centre for MathematicalAnalysis, Canberra, 1984.Jürgen Jost. Riemannian geometry and geometric analysis. Universitext.Springer-Verlag, Berlin, fourth edition, 2005.Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes,volume 288 of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences]. Springer-Verlag,Berlin, second edition, 2003.Wilfrid S. Kendall. Nonnegative Ricci curvature and the Brownian couplingproperty. Stochastics, 19(1-2):111–129, 1986.Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differentialgeometry. Vol. I. Wiley Classics Library. John Wiley & Sons Inc., NewYork, 1996. Reprint of the 1963 original, A Wiley-Interscience Publication.Hiroshi Kunita. Stochastic flows and stochastic differential equations,volume 24 of Cambridge Studies in Advanced Mathematics. CambridgeUniversity Press, Cambridge, 1990.[Lee97] John M. Lee. Riemannian manifolds, volume 176 of Graduate Textsin Mathematics. Springer-Verlag, New York, 1997. An introduction tocurvature.[Lot07][Mey81][Nor92][OW05]John Lott. Optimal transport and Ricci curvature for metric-measurespaces. In Surveys in differential geometry. Vol. XI, volume 11 of Surv.Differ. Geom., pages 229–257. Int. Press, Somerville, MA, 2007.P.-A. Meyer. Géométrie stochastique sans larmes. In Seminar on Probability,XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), volume850 of Lecture Notes in Math., pages 44–102. Springer, Berlin, 1981.J. R. Norris. A complete differential formalism for stochastic calculus inmanifolds. In Séminaire de Probabilités, XXVI, volume 1526 of LectureNotes in Math., pages 189–209. Springer, Berlin, 1992.Felix Otto and Michael Westdickenberg. Eulerian calculus for the contractionin the Wasserstein distance. SIAM J. Math. Anal., 37(4):1227–1255 (electronic), 2005.133
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Université de PoitiersTHÈSEpour o
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AbstractIn the first part of this t
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ivTABLE DES MATIÈRES3 Kendall-Cran
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8 CHAPITRE 1. INTRODUCTIONdifféren
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10 CHAPITRE 1. INTRODUCTIONdonne se
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12 CHAPITRE 1. INTRODUCTIONHamilton
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14 CHAPITRE 1. INTRODUCTIONx et ell
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16 CHAPITRE 1. INTRODUCTIONCette é
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18 CHAPITRE 1. INTRODUCTION- Exempl
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20 CHAPITRE 1. INTRODUCTION20
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22CHAPITRE 2. INTRODUCTION À L’A
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2. ÉQUATIONS DIFFÉRENTIELLES STOC
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2. ÉQUATIONS DIFFÉRENTIELLES STOC
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3. QUELQUES APPLICATIONS DU CALCUL
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Chapitre 3Brownian motion with resp
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for every smooth function f,is a lo
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For the solution U t of (1.1) we ge
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Remark : Recall that in the compact
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2 Local expression, evolution equat
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The last equality comes from Green
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Theorem 3.2 For every solution f(t,
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Consequently:d(df(T − t, .) X Tt
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Proof : The first remark after theo
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Remark : Hamilton gives a proof of
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Corollary 3.7 For χ(M) < 0, there
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We also have:D S,T −t dπ ˜// 0,
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Proof : By differentiation under x
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where we have used in the second eq
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[10] K. D. Elworthy and M. Yor. Con
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Chapter 4Some stochastic process wi
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We will just look at the smooth sol
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that is to say:d(Y T,it ) = − ∂
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2 Tightness, and first example on t
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proof : It is clear that F is smoot
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Proposition 2.6 Let g(t) be a famil
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v) ˜g(∞) is a metric such that (
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Then:for all ɛ > 0 , there exists
Page 81 and 82: Finally, we obtain:∂∂t | t=t 0
Page 83 and 84: where Ut 3 is the horizontal lift o
Page 85 and 86: √πWe can choose ɛ, ɛ 2 such th
Page 87 and 88: We will now show that the coupling
Page 89 and 90: HenceWe get:√√ n∑1 − ɛI t
Page 91 and 92: By uniqueness in law of such proces
Page 95 and 96: Chapter 5Horizontal diffusion in pa
Page 97 and 98: 2 M. ARNAUDON, A. K. COULIBALY, AND
Page 105 and 106: 10 M. ARNAUDON, A. K. COULIBALY, AN
Page 113 and 114: Chapter 6Compléments de calculs113
Page 115 and 116: d’avoir une famille de connexion,
Page 117 and 118: Dans le calcul de ligne 8 à ligne
Page 119 and 120: On utilise le fait que W (.) t est
Page 121 and 122: Chapter 7Appendix121121
Page 123 and 124: 774 M. Arnaudon et al. / C. R. Acad
Page 129 and 130: Bibliography[ABT02]Marc Arnaudon, R
Page 131: BIBLIOGRAPHY 131[DeT83][Dri92]Denni
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