Brownian Motion and Classical Potential Theory

Brownian Motion and Classical Potential Theory

1st Edition - September 28, 1978

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  • Author: Sidney Port
  • eBook ISBN: 9780323159081

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Brownian Motion and Classical Potential Theory is a six-chapter text that discusses the connection between Brownian motion and classical potential theory. The first three chapters of this book highlight the developing properties of Brownian motion with results from potential theory. The subsequent chapters are devoted to the harmonic and superharmonic functions, as well as the Dirichlet problem. These topics are followed by a discussion on the transient potential theory of Green potentials, with an emphasis on the Newtonian potentials, as well as the recurrent potential theory of logarithmic potentials. The last chapters deal with the application of Brownian motion to obtain the main theorems of classical potential theory. This book will be of value to physicists, chemists, and biologists.

Table of Contents

  • Preface

    Glossary of Notation

    Chapter 1 Brownian Motion as a Strong Markov Process

    1. Definition of Brownian Motion

    2. Monotone Class Theorem

    3. Markov Property

    4. Stopping Times

    5. Strong Markov Property

    Chapter 2 Hitting Times

    1. Auxiliary Analytical Results

    2. Elementary Properties of Hitting Times

    3. Regular Points

    4. Transition Operators for the Killed Process

    5. Properties of λ-Potentials

    6. Polar Sets

    7. Nonpolar Sets for Planar Brownian Motion

    8. Brownian Motion on an Interval

    Chapter 3 Potentials on the Whole Space

    1. Newtonian Potentials

    2. Asymptotic Behavior of Hitting Times

    3. Criteria for Regularity and Recurrence

    4. Logarithmic Potentials

    5. Linear Potentials

    Chapter 4 Harmonic Functions

    1. Basic Properties of Harmonic Functions

    2. Dirchlet Problem

    3. Poisson's Formula

    4. Nonnegative Harmonic Functions on an Open Ball

    5. Green Function

    6. Poisson's Equation

    7. Eigenfunction Expansion

    Chapter 5 Superharmonic and Excessive Functions

    1. Properties of Superharmonic and Excessive Functions

    2. Superharmonic Functions and Polar Sets

    3. Resolutivity

    4. Behavior along Brownian Motion Paths

    Chapter 6 Potential Theory

    1. Green Potentials

    2. Riesz Decomposition Theorem

    3. Balayage Problem

    4. Energy

    5. Equilibrium Problem

    6. Application to Electrostatics

    7. Logarithmic Potential Theory



Product details

  • No. of pages: 250
  • Language: English
  • Copyright: © Academic Press 1978
  • Published: September 28, 1978
  • Imprint: Academic Press
  • eBook ISBN: 9780323159081

About the Author

Sidney Port

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