Brownian Motion

Höfundar: Peter Mörters; Yuval Peres (Útgáfa: 1)
Brownian Motion

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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.

Nánar um bókina

Útgefandi
Cambridge University Press
ISBN
9781139930789
Print ISBN
9780521760188
Format
ePub
Útgáfa
1
Höfundar
Peter Mörters; Yuval Peres
Tungumál
English
Útgefið
2010-03-25
Prent takmörkun á líftíma
10
Prent takmörkun
5
Afritunar takmörkun
5

Kaflar

  • Cover
  • Half Title
  • Series Page
  • Title Page
  • Copyright
  • Contents
  • Preface
  • Frequently used notation
  • Motivation
  • 1. Brownian motion as a random function
  • 1.1 Paul Lévy’s construction of Brownian motion
  • 1.2 Continuity properties of Brownian motion
  • 1.3 Nondifferentiability of Brownian motion
  • 1.4 The Cameron–Martin theorem
  • Exercises
  • Notes and comments
  • 2. Brownian motion as a strong Markov process
  • 2.1 The Markov property and Blumenthal’s 0-1 law
  • 2.2 The strong Markov property and the reflection principle
  • 2.3 Markov processes derived from Brownian motion
  • 2.4 The martingale property of Brownian motion
  • Exercises
  • Notes and comments
  • 3. Harmonic functions, transience and recurrence
  • 3.1 Harmonic functions and the Dirichlet problem
  • 3.2 Recurrence and transience of Brownian motion
  • 3.3 Occupation measures and Green’s functions
  • 3.4 The harmonic measure
  • Exercises
  • Notes and comments
  • 4. Hausdorff dimension: Techniques and applications
  • 4.1 Minkowski and Hausdorff dimension
  • 4.2 The mass distribution principle
  • 4.3 The energy method
  • 4.4 Frostman’s lemma and capacity
  • Exercises
  • Notes and comments
  • 5. Brownian motion and random walk
  • 5.1 The law of the iterated logarithm
  • 5.2 Points of increase for random walk and Brownian motion
  • 5.3 Skorokhod embedding and Donsker’s invariance principle
  • 5.4 The arcsine laws for random walk and Brownian motion
  • 5.5 Pitman’s 2M − B theorem
  • Exercises
  • Notes and comments
  • 6. Brownian local time
  • 6.1 The local time at zero
  • 6.2 A random walk approach to the local time process
  • 6.3 The Ray–Knight theorem
  • 6.4 Brownian local time as a Hausdorff measure
  • Exercises
  • Notes and comments
  • 7. Stochastic integrals and applications
  • 7.1 Stochastic integrals with respect to Brownian motion
  • 7.2 Conformal invariance and winding numbers
  • 7.3 Tanaka’s formula and Brownian local time
  • 7.4 Feynman–Kac formulas and applications
  • Exercises
  • Notes and comments
  • 8. Potential theory of Brownian motion
  • 8.1 The Dirichlet problem revisited
  • 8.2 The equilibrium measure
  • 8.3 Polar sets and capacities
  • 8.4 Wiener’s test of regularity
  • Exercises
  • Notes and comments
  • 9. Intersections and self-intersections of Brownian paths
  • 9.1 Intersection of paths: Existence and Hausdorff dimension
  • 9.2 Intersection equivalence of Brownian motion and percolation limit sets
  • 9.3 Multiple points of Brownian paths
  • 9.4 Kaufman’s dimension doubling theorem
  • Exercises
  • Notes and comments
  • 10. Exceptional sets for Brownian motion
  • 10.1 The fast times of Brownian motion
  • 10.2 Packing dimension and limsup fractals
  • 10.3 Slow times of Brownian motion
  • 10.4 Cone points of planar Brownian motion
  • Exercises
  • Notes and comments
  • Appendix A: Further developments
  • 11. Stochastic Loewner evolution and planar Brownian motion
  • 11.1 Some subsets of planar Brownian paths
  • 11.2 Paths of stochastic Loewner evolution
  • 11.3 Special properties of SLE(6)
  • 11.4 Exponents of stochastic Loewner evolution
  • Notes and comments
  • Appendix B: Background and prerequisites
  • 12.1 Convergence of distributions
  • 12.2 Gaussian random variables
  • 12.3 Martingales in discrete time
  • 12.4 Trees and flows on trees
  • Hints and solutions for selected exercises
  • Selected open problems
  • Bibliography
  • Index