Brownian Motion and Potential Theory, Modern and Classical

Contents
Preface to the Second Edition
Preface to the First Edition
About the Authors
1 Introduction
	References
2 Martingales and Markov Processes
	2.1 Optional Sampling, Inequalities, and Convergence
	2.2 Continuous Parameter
	References
3 Brownian Motion and Ito Calculus
	3.1 The d-Dimensional Brownian Motion
	3.2 Strong Markov Property
	References
4 Semi-Groups of Operators, Potentials, and Diffusion Equations
	4.1 Semi-Groups
	4.2 Infinitesimal Generators
	4.3 Potential Operators
	References
5 Harmonic Functions, Dynkin, and Transforms
	5.1 Dynkin’s Formula
	5.2 Dirichlet Problem
	5.3 The Kelvin Transformation
	5.4 Boundary Limit Theorems of Fatou
	5.5 Spherical Harmonics
	References
6 Superharmonic Functions and Riesz Measures
	6.1 Superharmonic Functions
	6.2 Applications
	6.3 Riesz Measure
	6.4 The Continuity Principle
	6.5 The Dirichlet Problem Revisited
	References
7 Green Functions, Boundary Value Problems, and Kernels
	7.1 Green Functions for Bounded Open Sets
	7.2 Unbounded Open Subsets of R2
		7.2.1. Show that
		7.2.2. Show that
	7.3 Unbounded Open Sets (Continued)
	7.4 Examples
	7.5 The Green Function and Relative Transition
	References
8 Potential Theory, Capacity, Boundaries, Dirichlet Spaces, and Applications
	8.1 Some Potential Theoretic Principles
	8.2 Capacity
	8.3 Applications
	8.4 Balayage
	8.5 Dirichlet Spaces
	References
Appendix Kernels and More General Classes of Gaussian Processes
	A.1 Positive Definite Kernels
	A.2 Gaussian Processes
	A.3 Sigma-Finite Measure Spaces and Gaussian Processes
	References
Index