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Probabilistic Techniques in Analysis : Probability and Its Applications - Richard F. Bass

Probabilistic Techniques in Analysis

Probability and Its Applications

Hardcover Published: 16th December 1994
ISBN: 9780387943879
Number Of Pages: 392

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In recent years, there has been an upsurge of interest in using techniques drawn from probability to tackle problems in analysis. These applications arise in subjects such as potential theory, harmonic analysis, singular integrals, and the study of analytic functions. This book presents a modern survey of these methods at the level of a beginning Ph.D. student. Highlights of this book include the construction of the Martin boundary, probabilistic proofs of the boundary Harnack principle, Dahlberg's theorem, a probabilistic proof of Riesz' theorem on the Hilbert transform, and Makarov's theorems on the support of harmonic measure.
The author assumes that a reader has some background in basic real analysis, but the book includes proofs of all the results from probability theory and advanced analysis required. Each chapter concludes with exercises ranging from the routine to the difficult. In addition, there are included discussions of open problems and further avenues of research.

Preface
Probability
Brownian motionp. 8
Markov propertiesp. 15
Martingalesp. 28
Stochastic integralsp. 42
Stochastic calculusp. 54
Weak convergencep. 71
Potential Theory
The Dirichlet problemp. 83
Choquet capacitiesp. 97
Newtonian potentials and Green functionsp. 103
Transition densitiesp. 121
Newtonian capacityp. 135
Excessive functionsp. 147
Martin boundaryp. 155
Lipschitz Domains
The boundary Harnack principlep. 176
The Martin boundary for Lipschitz domainsp. 192
The conditional lifetime problemp. 198
Faton theoremsp. 207
Support of harmonic measurep. 213
Singular Integrals
Maximal functionsp. 230
Hilbert transformsp. 237
Riesz transformsp. 242
Littlewood-Paley functionsp. 259
Singular integral operators and Fourier multipliersp. 267
The space H[superscript 1]p. 278
Bounded mean oscillationp. 288
Analytic Functions
Conformal invariancep. 310
Range of analytic functionsp. 320
Boundary behavior of analytic functionsp. 326
Harmonic measurep. 337
The angular derivative problemp. 348
The corona problemp. 361
Bibliographyp. 375
Indexp. 387
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387943879
ISBN-10: 0387943870
Series: Probability and Its Applications
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 392
Published: 16th December 1994
Publisher: SPRINGER VERLAG GMBH
Country of Publication: US
Dimensions (cm): 24.28 x 16.33  x 2.67
Weight (kg): 0.76

Earn 339 Qantas Points
on this Book