Brownian motion /

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Bibliographic Details
Author / Creator:Mörters, Peter.
Imprint:Cambridge : Cambridge University Press, 2010.
Description:xii, 403 p. : ill. ; 26 cm.
Language:English
Series:Cambridge series in statistical and probabilistic mathematics ; [30]
Cambridge series on statistical and probabilistic mathematics ; 30.
Subject:Brownian motion processes.
Brownian motion processes.
Electronic books.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8209462
Hidden Bibliographic Details
Other authors / contributors:Peres, Y. (Yuval)
Schramm, Oded.
Werner, Wendelin, 1968-
ISBN:9780521760188 (Hardback)
0521760186
9780511743191 (e-book)
Notes:Includes bibliographical references (p. 386-399) and index.
Electronic reproduction. Palo Alto, Calif. : ebrary, 2010. Available via World Wide Web. Access may be limited to ebrary affiliated libraries.
Table of 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