http://dx.doi.org/10.1090/gsm/038 Selected Titles in This Series

38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993 This page intentionally left blank Stochastic Analysis on Manifolds This page intentionally left blank Stochasti c Analysi s on Manifold s

Elto n P . Hs u

Graduat e Studie s in Mathematic s Volume 38

American Mathematical Society Providence, Rhode Island WDE D X * Editorial Board Steven G. Krantz David Saltman (Chair) David Sattinger Ronald Stern

2000 Mathematics Subject Classification. Primary 58J65, 60J60, 60J65.

ABSTRACT. This book is intended for advanced graduate students and researchers who are familiar with both euclidean stochastic analysis and basic differential geometry. It begins with a brief review of stochastic differential equations on euclidean space. After presenting the basics of stochastic analysis on manifolds, we introduce Brownian motion on a Riemannian manifold and study the effect of curvature on its behavior. We then show how to apply Brownian motion to geometric problems and vice versa by several typical examples, including eigenvalue estimates and probabilistic proofs of the Gauss-Bonnet-Chern theorem and the Atiyah-Singer index theorem for Dirac operators. The book ends with an introduction to stochastic analysis on the path space over a Riemannian manifold, a topic of much current research interest.

Library of Congress Cataloging-in-Publication Data Hsu, Elton P., 1959- Stochastic analysis on manifolds / Elton P. Hsu. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 38) Includes bibliographical references and index. ISBN 0-8218-0802-8 (acid-free paper) 1. Stochastic differential equations. 2. Diffusion processes. 3. Differential geometry. I. Title. II. Series.

QA614.9.H78 2001 514'.74—dc21 2001046052

Copying ancl reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 In Memory of My Beloved Mother Zhu Pei-Ru (1926-1996)

Qualt mich Erinnerung, daB ich veriibet, So manche Tat, die dir das Herz betriibet? Das schone Herz, das mich so sehr geliebet?

Doch da bist du entgegen mir gekommen, Und ach! was da in deinem Aug geschwommen, Das war die siiBe, langgesuchte Liebe.

Heinrich Heine, An Meine Mutter. This page intentionally left blank Contents

Preface xiii Introduction 1 Chapter 1. Stochastic Differential Equations and Diffusions 5 §1.1. SDE on euclidean space 6 §1.2. SDE on manifolds 19 §1.3. Diffusion processes 24 Chapter 2. Basic Stochastic Differential Geometry 35 §2.1. Frame bundle and connection 36 §2.2. Tensor fields 41 §2.3. Horizontal lift and stochastic development 44 §2.4. Stochastic line integrals 51 §2.5. Martingales on manifolds 55 §2.6. Martingales on submanifolds 60 Chapter 3. Brownian Motion on Manifolds 71 §3.1. Laplace-Beltrami operator 72 §3.2. Brownian motion on manifolds 78 §3.3. Examples of Brownian motion 82 §3.4. Distance function 88 §3.5. Radial process 92 §3.6. An exit time estimate 98 Chapter 4. Brownian Motion and Heat Semigroup 101 X Contents

§4.1. Heat kernel as transition density function 102 §4.2. Stochastic completeness 107 §4.3. Co-property of the heat semigroup 112 §4.4. Recurrence and transience 117 §4.5. Comparison of heat kernels 127 Chapter 5. Short-time Asymptotics 129 §5.1. Short-time asymptotics: near points 130 §5.2. Varadhan's asymptotic relation 133 §5.3. Short-time asymptotics: distant points 137 §5.4. Brownian bridge 142 §5.5. Derivatives of the logarithmic heat kernel 147 Chapter 6. Further Applications 157 §6.1. Dirichlet problem at infinity 158 §6.2. Constant upper bound 166 §6.3. Vanishing upper bound 168 §6.4. Radially symmetric manifolds 172 §6.5. Coupling of Brownian motion 179 §6.6. Coupling and index form 182 §6.7. Eigenvalue estimates 186 Chapter 7. Brownian Motion and Index Theorems 191 §7.1. Weitzenbock formula 192 §7.2. Heat equation on differential forms 198 §7.3. Gauss-Bonnet-Chern formula 201 §7.4. and spin group 208 §7.5. Spin bundle and the 213 §7.6. Atiyah-Singer index theorem 217 §7.7. Brownian holonomy 222 Chapter 8. Analysis on Path Spaces 229 §8.1. Quasi-invariance of the Wiener measure 230 §8.2. Flat path space 234 §8.3. Gradient formulas 246 §8.4. Integration by parts in path space 250 §8.5. Martingale representation theorem 256 §8.6. Logarithmic Sobolev inequality and hypercontractivity 258 Contents XI

§8.7. Logarithmic Sobolev inequality on path space 263 Notes and Comments 267 General Notations 271 Bibliography 275 Index 279 This page intentionally left blank Preface

A large part of modern analysis is centered around the Laplace operator and its generalizations in various settings. On the other hand, under certain technical conditions one can show that the generator of a continuous strong Markov process must be a second order elliptic operator. These are the two facts that make the connection between probability and analysis seem natural. If we adopt the view that Brownian paths are the "characteristic lines" for the Laplace operator, then it is no longer surprising that solutions of many problems associated with the Laplace operator can be explicitly represented by Brownian motion, for it is a well known fact in the theory of partial differential equations that explicit solutions are possible if one knows characteristic lines. While analysts are interested in what happens in average, to a probabilist things usually happen at the path level. For these reasons probability theory, and Brownian motion in particular, has become a convenient language and useful tool in many areas of analysis. The purpose of this book is to explore this connection between Brownian motion and analysis in the area of differential geometry. Unlike many time-honored areas of mathematics, stochastic analysis has neither a well developed core nor a clearly defined boundary. For this reason the choice of the topics in this book reflects heavily my own interest in the subject; its scope is therefore much narrower than the title indicates. My purpose is to show how stochastic analysis and differential geometry can work together towards their mutual benefit. The book is written mainly from a probabilist's point of view and requires for its understanding a solid background in basic euclidean stochastic analysis. Although necessary geo• metric facts are reviewed throughout the book, a good knowledge of differen• tial geometry is assumed on the part of the reader. Because of its somewhat

xm XIV Preface unusual dual prerequisites, the book is best suited for highly motivated ad• vanced graduate students in either stochastic analysis or differential geom• etry and for researchers in these and related areas of mathematics. Notably absent from the book are a collection of exercises commonly associated with books of this kind, but throughout the book there are many proofs which are nothing but an invitation to test the reader's understanding of the topics under discussion. During the writing of this book, I have greatly benefited from several existant monographs on the subject; these include: • N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland/Kodansha (1989); • K. D. Elworthy: Stochastic Differential Equations on Manifolds, Cambridge University Press (1982); • M. Emery: Stochastic Calculus in Manifolds, Springer (1989); • P. Malliavin: Stochastic Analysis, Springer (1997). Overlaps with these works are not significant, and they and the more re• cent An Introduction to the Analysis of Paths on a Riemannian Manifold, American Mathematical Society (2000), by D. W. Stroock, are warmly rec• ommended to the reader. This book could not have been written without constant support from my wife, who has taken more than her fair share of family duties during its long gestation period. I would like to take this opportunity to thank Elena Kosygina, Tianhong Li, Banghe Li, and Mark A. Pinsky for reading early drafts of various parts of the book and for their valuable suggestions. I would also like to acknowledge many years of financial support through research grants from the National Science Foundation. Most of the book is based on the lectures I have delivered at various places during the late 1990s, notably at Academica Sinica in Beijing (1995), IAS/Park City Mathematics Institute in Princeton (1996), Institut Henri Poincare in Paris (1998), and Northwestern University in Evanston (1999), and I would like to thank my audiences for their comments and suggestions.

Elton P. Hsu Hinsdale, IL October, 2001. This page intentionally left blank General Notations

Notation Definition

3%(X) Borel cr-field of a metric space X

3§* Borel filtration of W(M) = {3St} B(x; R) (geodesic) ball of radius R centered at x DM Hodge-de Rham Laplacian on M (X, Y) co-variation of semimartingales X and Y (X) quadratic variation of X — (X, X) C°°(M) smooth functions on M Cx cutlocus of x o Stratonovich stochastic integral, X o dY = XdY + \d{X, Y) c(X) Clifford multiplication by X (in CHAPTER 7) d exterior differentiation ^M(^J y) distance between x and y on M D gradient operator on PQ(M) D Dirac operator (in CHAPTER 7) Dh Cameron-Martin vector field on P0{M) S dual of exterior differentiation AM Laplace-Beltrami operator on M = — (dS + 5d) A&(M) Bochner's horizontal Laplacian on &{M) — X^=i Hf A* dual operator of A e(uj) lifetime (explosion time) of a path u End(y) space of linear transforms on V exp0 exponential map based at o £P* filtration of cr-fields = {^t}

271 272 General Notations

Notation Definition

&(M) frame bundle of M ^^ filtration generated by process X = {^^} r(E) space of sections of vector bundle E rjfj Christoffel symbols r(f, g) r of / and g = L(fg) - fLg - gLf GL(d,M) set of real nonsingular (d x d) matrices Hi fundamental horizontal vector field on &{M) 3tf Cameron-Martin space i(X) interior product with X ix injectivity radius at x %K injectivity radius on K — min {ix : x G K} I (J, J) index form of a vector field J KM(X) set of sectional curvatures at a point x £ M L Orstein-Uhlenbeck operator (in CHAPTER 8) ^(d, /) space of (d x I) matrices M"!" transpose of a matrix M M one-point compactification of a manifold M —M U {<9M} \-\^ Cameron-Martin norm V connection and covariant differentiation V2/ Hessian of / H V G horizontal gradient of G = {HiG,..., HdG} 0(d) (d x d) orthogonal group o(d) (d x d) anti-symmetric matrices 0{M) orthonormal frame bundle of M ft curvature form Pf(A) Pfaffian of an anti-symmetric matrix A P0{M) space of paths on M starting from o with time length 1 Fx law of Brownian motion starting from x ^x,y-,t law °f Brownian bridge from x to y with time length t PM(t, %, y) heat kernel on a Riemannian manifold M tAjVf 2 {Pt} heat semigroups e / {^t} Ornstein-Uhlenbeck semigroups etL'2 II second fundamental form euclidean space of dimension N set of nonnegative real numbers = [0, oo), i?(X, Y)Z curvature tensor evaluated at X, Y, Z RICM(^) set of Ricci curvatures at a point x £ M General Notations 273

Notation Definition

Ricw Ricci transform at a frame u G &(M) y+(d) (d x d) symmetric positive definite matrices sn n-sphere Spin(d) Spin group y{M) spin bundle over a spin manifold M y&(M) Spin(d)-principal bundle over M X* horizontal lift of X e TM to the frame bundle &{M) TD first exit time of D — inf {t : Xt 0 D} 9t shift operator in a path space: (6toj)s = ^t+s TK first hitting time of K = inf {t : Xt G K} 6 scalarization of a tensor 6 Trace supertrace (in CHAPTER 7) TM tangent bundle of a manifold M TXM tangent space of M at x T*M cotangent space of M at x A?M space of p-forms at a point x G M W(M) space of paths on M with lifetimes This page intentionally left blank Bibliography

1. Aida, S. and El worthy, K. D., Differential calculus on path and loop spaces, I. Log- arthmic Sobolev inequalities on path spaces, C R. Acad. Sci. (Paris), 321, Serie I (1995), 97-102. 2. Azencott, R. et al., Geodesiques et diffusions en temps petit, Asterisque, 84—85 (1981). 3. Berger, M., Gauduchon, P. and Mazet, E., Le Spectre d'une Variete Riemannienne, Lect. Notes in Math., 194, Springer-Verlag (1971). 4. Bishop, R. L. and Crittenden, R. J., Geometry of Manifolds, Academic Press (1964). 5. Bismut, J.M., The Atiyah-Singer Theorems: A Probabilistic Approach, J. of Func. Anal, Part I: 57 (1984), 56-99; Part II: 57 (1984), 329-348. 6. Bismut, J-M., Large Deviations and the Malliavin Calculus, Birkhauser (1984). 7. Capitaine, M., Hsu, E. P., and Ledoux, M., Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces, Electronic Communications m Probability, 2 Paper no. 7 (1997), 71-81. 8. Chavel, I. Eigenvalues in Riemannian Geometry, Academic Press (1984). 9. Cheeger, J. and Ebin, D. G., Comparison Theorems in Differential Geometry, North- Holland/Kodansha (1975). 10. Chen, M. F. and Wang, F. Y., Application of coupling method to the first eigenvalue on manifold, Science in China (A), 37, no. 1 (1994), 1-14. 11. Chung, K. L., Lectures from Markov Processes to Brownian Motion, Grund. Math. Wiss. 249, Springer-Verlag (1982). 12. Copson, E. T., Asymptotic Expansions, Cambridge University Press (1965). 13. Courant, R. and Hilbert, D., Methods of Mathematical Physics, V.l, Interscience, New York (1962). 14. Cranston, M., Gradient estimates on manifolds using coupling, J. of Func. Anal., 99 (1991), 110-124. 15. Deuschel, D. J.-D. and Stroock, D. W., Large Deviations, Academic Press (1989). 16. Debiard, A., Gaveau, B., and Mazet, E., Theoremes de comparaison en geometrie riemannienne, Publ. RIMS, Kyoto Univ., 12 (1976), 391-425.

275 276 Bibliography

17. Do Carmo, M. P., Riemannian Geometry, 2nd edition, Birkhauser (1993). 18. Donnelly, H. and Li, P., Lower bounds for the eigenvalues of Riemannian manifolds, Michigan Math. J., 29 (1982), 149-161. 19. Dodziuk, J., Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Math. J., 32, No. 5 (1983), 703- 716. 20. Driver, B., A Cameron-Martin type of quasiinvariance for the Brownian motion on a compact maniofld, J. of Func. Anal, 110 (1992), 237-376. 21. Elworthy, K. D., Stochastic Differential Equations on Manifolds, Cambridge Univer• sity Press (1982). 22. Elworthy, K. D. and Truman, A., The diffusion equation and classical mechanics: an elementary formula, in Stochastic Processes in Quantum Physics, ed. S. Albeverio et al., Lect. Notes in Physics 173, Springer-Verlag (1982), 136-146. 23. Elworthy, K. D., Geometric aspects of diffusions on manifolds, in Ecole d'Ete de Probabilites de Saint-Flour XV-XVII, 1985-1987, ed. by P. Diaconis et al., Lect. Notes in Math., 1382, Springer-Verlag (1988), 227-425. 24. Emery, Michel, Stochastic Calculus in Manifolds, Springer-Verlag (1989). 25. Emery, Michel, Martingales continues dans les varietes differentiables, Lectures on Probability Theory and Statistics, Ecole d'Ete de Probabilites de Saint-Flour XXVIII, 1998, Lect. Notes in Math., 1738 (2000), 1-84. 26. Enchev, O. and Stroock, D. W., Towards a Riemannian geometry on the path space over a Riemannian manifold, J. of Func. Anal, 134 (1995), 329-416. 27. Fang, S., Un inequality du type Poincare sur un espace de chemins, C. R. Acad. Sci. (Paris), 318, Serie I (1994), 257-260. 28. Friedrich, T., Dirac Operators in Riemannian Geometry, Graduate Studies in Math• ematics, Vol. 25, Amer. Math. Soc. (2000). 29. Fukushima, M., Dirichlet Forms and Markov Processes, North-Holland/Kodansha (1975). 30. Gallot, S., Hulin, D., and Lafontaine, J., Riemannian Geometry, Springer-Verlag (1990). 31. Gilkey, P., Invariance Theory, the Heat Equation, and the Atiyah-Singer Index The• orem, Publish or Perish, Inc., Wilmington, DE (1984). 32. Grigor'yan, A. A., On stochastic completeness of manifolds, Soviet Math. Dokl, 34 (1987), 310-313. 33. Grigor'yan, A. A., Analytic and geometric background of recurrence and non- explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (New Series), 36, No. 2 (1999), 135-249. 34. Gross, L., Logarithmic Sobolev inequalities, Amer. J. of Math., 97 (1975), 1061- 1083. 35. Gross, L., Logarithmic Sobolev inequalities and contractivity properties of semi• groups, in Dirichlet Forms, ed. by E. Fabes et al. Lect. Notes in Math., 1563, Springer-Verlag (1993), 54-82. 36. Hackenbroch, W. and Thalmaier, A., Stochastische Analysis, B. G. Teubner, Stuttgart (1994). 37. Hsu, P. and March, P., The limiting angle of certain Riemannian Brownian motions, Comm. Pure Appl. Math., 38, 755-768 (1985). 38. Hsu, P., Heat semigroup on a complete Riemannain manifold, Ann. of Probab., 17, no. 3 (1989), 1248-1254. Bibliography 277

39. Hsu, P., Brownian bridges on Riemannian manifolds, Probab. Theory and Rel. Fields, 84 (1990), 103-118. 40. Hsu, P., Heat kernel on noncomplete manifolds, Indiana Math. J., 39, no.2 (1990), 431-442. 41. Hsu, E. P., Quasiinvariance of the Wiener measure and integration by parts in the path space over a compact Riemannian manifold, J. of Func. Anal., 134 (1995) 417-450. 42. Hsu, E. P., Une inegalite logarithmique de Sobolev sur l'espace de chemins d'une variete riemannienne, C. R. Acad. Sci. (Paris), 320, Serie I (1995), 1009-1012. 43. Hsu, E. P., Stochastic local Gauss-Bonnet-Chern theorem, Journal of Theo. Probab., 10, no. 4 (1997), 819-834. 44. Hsu, E. P., Logarithmic Sobolev inequalities on the path space of a Riemannian manifold, Comm. Math. Phy., 189 (1997), 9-16. 45. Hsu, E. P., Analysis on path and loop spaces, in Probability Theory and Applications, IAS/Park City Mathematics Series, Vol. 6, edited by E. P. Hsu and S. R. S. Varadhan, AMS (1999), 277-347. 46. Hsu, E. P., Estimates of the derivatives of the heat kernel, Proceedings of AMS, 127 (1999), 3739-3744. 47. Ichihara, K., Curvature, geodesies and the Brownian otion on a Riemannian manifold I, Nagoya Math. J., 87 (1982), 101-114. 48. Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Pro• cesses, 2nd edition, North-Holland/Kodansha (1989). 49. Janson, S., Gaussian Hilbert Spaces, Cambridge University Press (1997). 50. Jost, J., Riemannian Geometry and Geometric Analysis, Springer-Verlag (1995). 51. Kendall, W. S., Nonnegative Ricci curvature and the Brownian coupling property, Stochastics 19 (1986), 111-129. 52. Kendall, W. S., The radial part of Brownian motion on a manifold: a semimartingale property, Ann. Probab., 15, no. 4 (1987), 1491-1500. 53. Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Vol. I, Inter- science Publishers, New York (1963). 54. Leclercq, E., The asymptotic Dirichlet problem with respect to an elliptic operator on a Cartan-Hadamard manifold with unbounded curvatures, C. R. Acad. Sci., 325, Serie I (1997), 857-862. 55. Li, P., Lecture Notes on Differential Geometry, Notes of the Series of Lectures Held at the Seoul National University, Vol. 6, Seoul National University (1993). 56. McKean, H.P. and Singer, I.M., Curvature and the eigenvalues of Laplacian, J. of Diff. Geo., 1 (1967), 43-69. 57. Malliavin, P., Stochastic Analysis, Springer-Verlag (1997). 58. March, P., Brownian motion and harmonic functions on rotationally symmetric man• ifolds, Ann. Probab., 14, No. 4 (1986), 793-804. 59. Molchanov, S. A., Diffusion processes and Riemannian geometry, Russian Math. Surveys, 30, No. 1 (1975), 1-63. 60. Norris, J.R., Path integral formulae for heat kernels and their derivatives, Probab. Theory Relat. Fields, 94 (1993), 525-541. 61. Patodi, V.K., Curvature and the eigenforms of the Laplace operator, J. of Diff. Geo., 5 (1971), 233-249. 278 Bibliography

62. Phillips, R. S. and Sarason, L., Elliptic parabolic equations of second order, J. Math. and Mech., 17, 891-917 (1967). 63. de Rham, G., Riemannian Manifolds, Springer-Verlag (1984). 64. Schoen, R. and Yau, S.-T. Lectures on Differential Geometry, International Press, Cambridge, MA (1994). 65. Schwartz, Laurent, Semimartingales and Stochastic Calculus on Manifolds, Les Presses de l'Universite de Montreal, Montreal, Canada (1984). 66. Sheu, S.-Y., Some estimates of the transition density function of a nondegenerate diffusion Markov process, Ann. Probab., 19, no. 2 (1991), 538-561. 67. Shiga, T. and Watanabe, S., Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahr. verw. Geb., 27 (1973), 37-46. 68. Stroock, D. W. and Varadhan, S. R. S., Multidimensional Diffusion Processes, Springer-Verlag (1979). 69. Stroock, D. W., An Introduction to the Analysis of Paths on a Riemannian Manifold, Mathematical Surveys and Monographs, Vol. 74, Amer. Math. Soc. (2000). 70. Varopoulos, N., Potential theory and diffusion on Riemannian manifolds, in Confer• ence on Harminic Analysis in Honor of Antoni Zygmund, Vol. II, Wadsworth Math. Ser., Wadsworth, Belmont, CA (1983), 821-837 71. Warner, F. W., Foundations of Differential Geometry and Lie Groups, Springer (1983). 72. Zhong, J. Q. and Yang, H. C, Estimates of the first eigenvalue of a compact Rie• mannian manifold, Scientia Sinica, 27, no. 12 (1984), 1251-1265. Index

Co-property, 113 Chapman-Kolmogorov equation, 103, 105 L-diffusion, 24 Chern character, 218 A-genus, 218 Christoffel symbol, 36, 73 V-martingale, 55 Clark-Ocone formula, 256 Clifford algebra, 209 additive functional, 96 Clifford multiplication, 213 adjoint operator, 239 closed operator, 239 angular process, 88 cohomology group, 201 anti-development, 40, 45 cone topology, 158 associated bundle, 38 connection, 36 Atiyah-Singer index theorem, 217 Levi-Civita, 43, 73 product, 68 Bessel process, 170 torsion-free, 43 Bismut's formula, 248 connection form, 149 Bochner's horizontal Laplacian, 193 continuous additive functional, 96 bridge space, 142 convex hypersurface, 62 Brownian bridge, 18, 142 coordinate process, 102 Brownian holonomy, 200 core, 239 Brownian motion, 78 cotangent space, 41 ^•-, 16 coupling time, 179 horizontal, 82 covariant differentiation, 36 in local coordinates, 86 covariant Laplacian, 192 on circle, 82 curvature on manifold, 65, 78 Ricci, 89 on radially symmetric manifold, 84 scalar, 219 on sphere, 83 sectional, 89 bundle curvature form, 149, 197 associated, 38 curvature tensor, 89 frame, 37 cut locus, 88 orthonormal frame, 40 cylinder function, 235 principal, 37 spin, 213 development, 40 differential form, 193 Cameron-Martin space, 230 diffusion coefficient, 6 canonical 1-form, 149 diffusion measure, 24 Cartan-Hadamard manifold, 111, 128, 158 diffusion procress, 24

279 280 Index

Dirac operator, 215 Hodge-de Rham theory, 201 Dirichlet form, 245 Hopf-Rinow theorem, 88 Dirichlet heat kernel, 102 horizontal Brownian motion, 82 Dirichlet problem, 118 horizontal curve, 38 Dirichlet problem at infinity, 159 horizontal gradient, 146 divergence, 74 horizontal Laplacian, 75, 192 Doob's inequality, 8, 240 horizontal lift dual operator, 239 of curve, 38 of semimartingale, 45 Eells-Elworthy-Malliavin construction, 2, 35 of vector, 38 embedding theorem horizontal semimartingale, 45 Nash, 76 horizontal vector, 38 Whitney, 21 hyperbolic manifold, 117 Euler form, 208 hypercontractivity, 260 exhaustion, 11, 104 explosion time, 5, 11 index form, 91, 183 exponential map, 58 index lemma, 91, 187 exponential martingale, 17, 145 injectivity radius, 89 exterior differentiation, 193 integration by parts formula, 236, 252 exterior product, 193 interior product, 194 isometric embedding, 76 Feller property, 113 Ito's formula, 7 Feynman-Kac formula, 198 in Stratonovich form, 19 filtered probability space, 6 first Bianchi identity, 217 Jacobi equation, 90 first exit time, 104 Jacobi field, 91, 182 first variation formula, 183 frame, 37 Kendall-Cranston coupling, 186 frame bundle, 37 Kolmogorov continuity criterion, 143 orthonormal, 40 Kolomogorov's extension theorem, 143 fundamental horizontal vector field, 39 fundamental solution, 105 Laplace-Beltrami operator, 74 fundamental vertical vector field, 149 Laplacian covariant, 192 geodesic, 37 Hodge-de Rham, 193 geodesically complete, 88 Laplacian comparison theorem, 90 Girsanov's theorem, 146, 230 law of the iterated logarithm, 167 gradient, 236 Levi-Civita connection, 43, 73 Green function, 121 Levy's criterion, 28 Gross' equivalence theorem, 261 Lichnerowicz formula, 215 Gross' logarithmic Sobolev inequality, 258 Lichnerowicz theorem, 188 Lie bracket, 43 half-spin representation, 209 lifetime, 11 harmonic forms, 201 Lipschitz condition, 6 heat equation, 198 local convergence theorem, 59 heat kernel, 105 local Gauss-Bonnet-Chern theorem, 206 Dirichlet, 102 logarithmic Sobolev inequality, 258 minimal, 105 loop space, 142 on euclidean space, 102 on forms, 200 manifold on space forms, 132 Cartan-Hadamard, 158 heat semigroup, 112, 113 geodesically complete, 88 Hermite polynomial, 243 Riemannian, 40, 72 hermitian vector bundle, 215 spin, 213 Hessian, 43 stochastically complete, 107 Hodge decomposition, 203 martingale Hodge-de Rham Laplacian, 193 exponential, 17, 145 Index 281

nonconfluence, 65 second variation formula, 91, 184 on manifold, 55 sectional curvature, 89 on submanifold, 60 semimartingale, 6 martingale problem, 24 horizontal, 45 martingale representation theorem, 85, 257 on manifold, 19 maximum principle, 119 up to a stopping time, 12 mean curvature, 62 shift operator, 33 minimal heat kernel, 105 solder form, 53 minimal submanifold, 62 space multiplicative functional, 198 bridge, 142 Myer's theorem, 188 loop, 142 path, 11 Nash's embedding theorem, 76 spectral gap, 179, 245 Nelson's hypercontractivity theorem, 262 sphere at infinity, 158 nonconfluence of martingales, 65 spin bundle, 213 normal coordinates, 58 spin group, 209 spin manifold, 213 one-point compactification, 11 spin representation, 209 Ornstein-Uhlenbeck operator, 241 , 213 Ornstein-Uhlenbeck process, 17 stochastic area formula, 221 orthonormal frame bundle, 40 stochastic differential equation on euclidean space, 6 parabolic manifold, 117 on manifold, 20 parallel transport, 37, 38 stochastic differential geometry, 35 path space, 11 stochastic line integral, 52 Patodi's local index theorem, 191 stochastic parallel transport, 50 Pfaffian, 208, 210 stochastically complete, 107 Picard's iteration, 9 Stratonovich formulation, 19 Poincare inequality, 263 strong Markov property, 31 principal bundle, 37 structure equation, 149 probability space, 6 submanifold filtered, 6 minimal, 62 process totally geodesic, 61 Bessel, 170 supertrace, 201, 211 coordinate, 102 Ornstein-Uhlenbeck, 17 tangent bundle, 36 radial, 88 tangent space, 36 product connection, 68 tensor field, 41 torsion, 43 quadratic variation, 53 torsion-free connection, 43 quasi-invariance, 230 totally geodesic submanifold, 61 transience, 117 radial process, 71, 88, 92 radially symmetric manifold, 172 uniqueness in law, 15 Rauch comparison theorem, 165 recurrence, 117 Varadhan's asmptotic relation, 133 regular conditional probability, 31 vertical vector, 38 Ricci curvature, 89 volume measure, 73 Riemannian manifold, 40, 72 Riemannian metric, 72 warped product, 85 rolling without slipping, 40 weak uniqueness, 15 Weitzenbock formula, 196 , 90, 217, 219 Whitney's embedding theorem, 21 scalarization, 42 Wiener chaos decomposition, 244 Schwartz test function, 243 Wiener measure, 78 second fundamental form, 61 second order elliptic operator, 24 Yamada's theorem, 97