Brownian Motion (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 30)

This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes. Cover......Page 1 Half-title......Page 3 Series-title......Page 4 Title......Page 5 Copyright......Page 6 Contents......Page 7 Preface......Page 10 Vectors, functions, and measures:......Page 12 Stopping times:......Page 13 Sets and processes associated with Brownian motion:......Page 14 Motivation......Page 15 1.1.1 Definition of Brownian motion......Page 21 1.1.2 Paul Lévy’s construction of Brownian motion......Page 23 1.1.3 Simple invariance properties of Brownian motion......Page 26 1.2 Continuity properties of Brownian motion......Page 28 1.3 Nondifferentiability of Brownian motion......Page 32 1.4 The Cameron–Martin theorem......Page 38 Exercises......Page 44 Notes and comments......Page 47 2.1 The Markov property and Blumenthal’s 0-1 law......Page 50 2.2 The strong Markov property and the reflection principle......Page 54 2.2.1 The reflection principle......Page 58 2.2.2 The area of planar Brownian motion......Page 59 2.3 Markov processes derived from Brownian motion......Page 62 2.4 The martingale property of Brownian motion......Page 67 Exercises......Page 73 Notes and comments......Page 77 3.1 Harmonic functions and the Dirichlet problem......Page 79 3.2 Recurrence and transience of Brownian motion......Page 85 3.3 Occupation measures and Green’s functions......Page 90 3.4 The harmonic measure......Page 98 Exercises......Page 105 Notes and comments......Page 108 4.1.1 The Minkowski dimension......Page 110 4.1.2 The Hausdorff dimension......Page 112 4.1.3 Upper bounds on the Hausdorff dimension......Page 115 4.2 The mass distribution principle......Page 119 4.3 The energy method......Page 122 4.4 Frostman’s lemma and capacity......Page 125 Exercises......Page 129 Notes and comments......Page 130 5.1 The law of the iterated logarithm......Page 132 5.2 Points of increase for random walk and Brownian motion......Page 137 5.3.1 The Dubins’ embedding theorem......Page 141 5.3.2 The Azéma–Yor embedding theorem......Page 143 5.3.3 The Donsker invariance principle......Page 145 5.4 The arcsine laws for random walk and Brownian motion......Page 149 5.5 Pitman’s 2M - B theorem......Page 154 Exercises......Page 160 Notes and comments......Page 163 6.1 The local time at zero......Page 167 6.2 A random walk approach to the local time process......Page 179 6.3 The Ray–Knight theorem......Page 184 6.4 Brownian local time as a Hausdorff measure......Page 192 Exercises......Page 200 Notes and comments......Page 201 7.1.1 Construction of the stochastic integral......Page 204 7.1.2 Itô’s formula......Page 209 7.2 Conformal invariance and winding numbers......Page 215 7.3 Tanaka’s formula and Brownian local time......Page 223 7.4 Feynman–Kac formulas and applications......Page 227 Exercises......Page 234 Notes and comments......Page 236 8.1 The Dirichlet problem revisited......Page 238 8.2 The equilibrium measure......Page 241 8.3 Polar sets and capacities......Page 248 8.4 Wiener’s test of regularity......Page 262 Exercises......Page 265 Notes and comments......Page 267 9.1.1 Existence of intersections......Page 269 9.1.2 Stochastic co-dimension and percolation limit sets......Page 272 9.1.3 Hausdorff dimension of intersections......Page 274 9.2 Intersection equivalence of Brownian motion and percolation limit sets......Page 277 9.3 Multiple points of Brownian paths......Page 286 9.4 Kaufman’s dimension doubling theorem......Page 293 Exercises......Page 299 Notes and comments......Page 301 10.1 The fast times of Brownian motion......Page 304 10.2 Packing dimension and limsup fractals......Page 312 10.3 Slow times of Brownian motion......Page 321 10.4 Cone points of planar Brownian motion......Page 326 Exercises......Page 336 Notes and comments......Page 338 11.1.1 The questions......Page 341 11.1.2 Reformulation in terms of Brownian hulls......Page 343 11.1.3 An alternative characterisation of Brownian hulls......Page 344 11.2.1 Heuristic description......Page 345 11.2.2 Loewner’s equation......Page 346 11.2.3 The loop-erased random walk......Page 347 11.2.5 Critical percolation and SLE(6)......Page 350 11.3 Special properties of SLE(6)......Page 353 11.4.1 A radial computation......Page 354 11.4.2 Consequences......Page 356 11.4.3 From exponents to dimensions......Page 357 12.1 Convergence of distributions......Page 360 12.2 Gaussian random variables......Page 363 Notes and comments......Page 358 12.3 Martingales in discrete time......Page 365 12.4 Trees and flows on trees......Page 372 Hints and solutions for selected exercises......Page 375 Selected open problems......Page 397 Bibliography......Page 400 Index......Page 414 This Textbook Offers A Broad And Deep Exposition Of Brownian Motion. Extensively Class Tested, It Leads The Reader From The Basics To The Latest Research In The Area. Starting With The Construction Of Brownian Motion, The Book Then Proceeds To Sample Path Properties Such As Continuity And Nowhere Differentiability. Notions Of Fractal Dimension Are Introduced Early And Are Used Throughout The Book To Describe Fine Properties Of Brownian Paths. The Relation Of Brownian Motion And Random Walk Is Explored From Several Viewpoints, Including A Development Of The Theory Of Brownian Local Times From Random Walk Embeddings. Stochastic Integration Is Introduced As A Tool, And An Accessible Treatment Of The Potential Theory Of Brownian Motion Clears The Path For An Extensive Treatment Of Intersections Of Brownian Paths. An Investigation Of Exceptional Points On The Brownian Path And An Appendix On Sle Processes, By Oded Schramm And Wendelin Werner, Lead Directly To Recent Research Themes.--jacket. Preface -- Frequently Used Notation -- Motivation -- Brownian Motion As A Random Function -- Brownian Motion As A Strong Markov Process -- Harmonic Functions, Transience And Recurrence -- Hausdorff Dimension: Techniques And Applications -- Brownian Motion And Random Walk -- Brownian Local Time -- Stochastic Integrals And Applications -- Potential Theory Of Brownian Motion -- Intersections And Self-intersections Of Brownian Paths -- Exceptional Sets For Brownian Motion -- Appendix A: Stochastic Loewner And Planar Brownian Motion -- Appendix B: Background And Prerequisites. Peter Mörters And Yuval Peres ; With An Appendix By Oded Schramm And Wendelin Werner. Includes Bibliographical References (p. 386-399) And Index. "This eagerly awaited textbook offers a broad and deep exposition of Brownian motion. Extensively class tested, it leads the reader from the basics to the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties such as continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool, and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes."-- taken from back cover