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Classical Potential Theory and Its Probabilistic Counterpart : Classics in Mathematics - Joseph L. Doob

Classical Potential Theory and Its Probabilistic Counterpart

Classics in Mathematics

Paperback

Published: 12th January 2001
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From the reviews: "This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner."
M. Brelot in Metrika (1986)

From the reviews:

"In the early 1920's, Norbert Wiener wrote significant papers on the Dirichlet problem and on Brownian motion. Since then there has been enormous activity in potential theory and stochastic processes, in which both subjects have reached a high degree of polish and their close relation has been discovered. Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1." G.E.H. Reuter in Short Book Reviews (1985)

"This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fullfilled in a masterly manner." Metrika (1986)

"It is good news that Doob's monumental book is now available at a very reasonable price. The impressive volume (846 pages!) is still the only book concentrating on a thorough presentation of the potential theory of the Laplace operator ... . The material in the chapters on conditional Brownian motion and Brownian motion on the Martin space cannot easily be found in that depth elsewhere. A long appendix on various topics (more than 50 pages) and many historical notes complete this great 'encyclopedia'." (Wolfhard Hansen, Zentralblatt MATH, Vol. 990 (15), 2002)

Introduction
Notation and Conventions
Classical and Parabolic Potential Theory
Introduction to the Mathematical Background of Classical Potential Theoryp. 3
Basic Properties of Harmonic, Subharmonic, and Superharmonic Functionsp. 14
Infima of Families of Superharmonic Functionsp. 35
Potentials on Special Open Setsp. 45
Polar Sets and Their Applicationsp. 57
The Fundamental Convergence Theorem and the Reduction Operationp. 70
Green Functionsp. 85
The Dirichlet Problem for Relative Harmonic Functionsp. 98
Lattices and Related Classes of Functionsp. 141
The Sweeping Operationp. 155
The Fine Topologyp. 166
The Martin Boundaryp. 195
Classical Energy and Capacityp. 226
One-Dimensional Potential Theoryp. 256
Parabolic Potential Theory: Basic Factsp. 262
Subparabolic, Superparabolic, and Parabolic Functions on a Slabp. 285
Parabolic Potential Theory (Continued)p. 295
The Parabolic Dirichlet Problem, Sweeping, and Exceptional Setsp. 329
The Martin Boundary in the Parabolic Contextp. 363
Probabilistic Counterpart of Part I
Fundamental Concepts of Probabilityp. 387
Optional Times and Associated Conceptsp. 413
Elements of Martingale Theoryp. 432
Basic Properties of Continuous Parameter Supermartingalesp. 463
Lattices and Related Classes of Stochastic Processesp. 520
Markov Processesp. 539
Brownian Motionp. 570
The Ito Integralp. 599
Brownian Motion and Martingale Theoryp. 627
Conditional Brownian Motionp. 668
Lattices in Classical Potential Theory and Martingale Theoryp. 705
Brownian Motion and the PWB Methodp. 719
Brownian Motion on the Martin Spacep. 727
App. I: Analytic Setsp. 741
Capacity Theoryp. 750
Lattice Theoryp. 758
Lattice Theoretic Concepts in Measure Theoryp. 767
Uniform Integrabilityp. 779
Kernels and Transition Functionsp. 781
Integral Limit Theoremsp. 785
Lower Semicontinuous Functionsp. 791
Historical Notesp. 793
Bibliographyp. 819
Notation Indexp. 827
Indexp. 829
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540412069
ISBN-10: 3540412069
Series: Classics in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 1551
Published: 12th January 2001
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.72 x 16.0  x 4.85
Weight (kg): 1.27