Brownian Motion and Classical Potential TheoryBrownian 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. |
Contents
| 1 | |
| 16 | |
Chapter 3 Potentials on the Whole Space | 53 |
Chapter 4 Harmonic Functions | 85 |
Chapter 5 Superharmonic and Excessive Functions | 128 |
Chapter 6 Potential Theory | 157 |
| 229 | |
| 233 | |
Probability and Mathematical Statistics | 237 |
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Common terms and phrases
a.s. Px B₁ boundary value zero Brownian motion Brownian motion starting CD(B charge polar sets Choose compact open subset compact subset compact support completes the proof concentrated Consequently continuous function converges completely converges strongly converges vaguely D₁ defined denote Dirichlet problem dominated convergence theorem equilibrium measure f₁ Fatou's lemma finite measure follows easily follows from Proposition follows from Theorem function f fundamental identity Gµ greatest harmonic minorant Green potential Greenian harmonic function hence Let f let ƒ Let µ logarithmic potentials lower semicontinuous Markov property measure µ nonempty nonnegative superharmonic function nonpolar open set open subset P(TB polar sets positive integer Px(TB Radon measure relatively compact open result follows strong Markov property subset of Rd superharmonic function uniquely X(TB xe Rd