Brownian motion

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Brownian motion / Peter Mörters and Yuval Peres ; with an appendix by Oded Schramm and Wendelin Werner

資料種別:

電子ブック

出版情報:

[Cambridge] : Cambridge University Press, 2012

形態:

1 online resource (xii, 403 p.)

シリーズ名:

Cambridge series on statistical and probabilistic mathematics <BA30272976>

著者名:

ISBN:

9780511750489 [051175048X] (: electronic bk)

書誌ID:

TT00015870

子書誌情報

フルテキスト

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書誌詳細

注記:

Bibliography: p. 386-399
Description based on online resource(viewed on July 16, 2019)
Includes index
原資料の出版事項: Cambridge : Cambridge University Press , 2010

主題:

Brownian motion processes

言語:

英語

目次情報:

Preface

Frequently used notation

Motivation

Brownian motion as a random function / 1：

Paul Lévy's construction of Brownian motion / 1.1：

Continuity properties of Brownian motion / 1.2：

Nondifferentiability of Brownian motion / 1.3：

The Cameron-Martin theorem / 1.4：

Exercises

Notes and comments

Brownian motion as a strong Markov process / 2：

The Markov property and Blumenthal's 0-1 law / 2.1：

The strong Markov property and the reflection principle / 2.2：

Markov processes derived from Brownian motion / 2.3：

The martingale property of Brownian motion / 2.4：

Harmonic functions, transience and recurrence / 3：

Harmonic functions and the Dirichlet problem / 3.1：

Recurrence and transience of Brownian motion / 3.2：

Occupation measures and Green's functions / 3.3：

The harmonic measure / 3.4：

Hausdorff dimension: Techniques and applications / 4：

Minkowski and Hausdorff dimension / 4.1：

The mass distribution principle / 4.2：

The energy method / 4.3：

Frostman's lemma and capacity / 4.4：

Brownian motion and random walk / 5：

The law of the iterated logarithm / 5.1：

Points of increase for random walk and Brownian motion / 5.2：

Skorokhod embedding and Donsker's invariance principle / 5.3：

The arcsine laws for random walk and Brownian motion / 5.4：

Pitman's 2M - B theorem / 5.5：

Brownian local time / 6：

The local time at zero / 6.1：

A random walk approach to the local time process / 6.2：

The Ray-Knight theorem / 6.3：

Brownian local time as a Hausdorff measure / 6.4：

Stochastic integrals and applications / 7：

Stochastic integrals with respect to Brownian motion / 7.1：

Conformal invariance and winding numbers / 7.2：

Tanaka's formula and Brownian local time / 7.3：

Feynman-Kac formulas and applications / 7.4：

Potential theory of Brownian motion / 8：

The Dirichlet problem revisited / 8.1：

The equilibrium measure / 8.2：

Polar sets and capacities / 8.3：

Wiener's test of regularity / 8.4：

Intersections and self-intersections of Brownian paths / 9：

Intersection of paths: Existence and Hausdorff dimension / 9.1：

Intersection equivalence of Brownian motion and percolation limit sets / 9.2：

Multiple points of Brownian paths / 9.3：

Kaufman's dimension doubling theorem / 9.4：

Exceptional sets for Brownian motion / 10：

The fast times of Brownian motion / 10.1：

Packing dimension and limsup fractals / 10.2：

Slow times of Brownian motion / 10.3：

Cone points of planar Brownian motion / 10.4：

Further developments / Appendix A：

Stochastic Loewner evolution and planar Brownian motion / Oded Schramm ; Wendelin Werner11：

Some subsets of planar Brownian paths / 11.1：

Paths of stochastic Loewner evolution / 11.2：

Special properties of SLE(6) / 11.3：

Exponents of stochastic Loewner evolution / 11.4：

Background and prerequisites / Appendix B：

Convergence of distributions / 12.1：

Gaussian random variables / 12.2：

Martingales in discrete time / 12.3：

Trees and flows on trees / 12.4：

Hints and solutions for selected exercises

Selected open problems

Bibliography

Index

Hausdorff dimension: techniques and applications

Stochastic Loewner evolution and its applications to planar Brownian motion

References

Preface

Frequently used notation

Motivation

Brownian motion as a random function / 1：

Paul Lévy's construction of Brownian motion / 1.1：

Continuity properties of Brownian motion / 1.2：

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