Classical Potential Theory and Its Probabilistic Counterpart Book

	Introduction
	Notation and Conventions
	Classical and Parabolic Potential Theory
	Introduction to the Mathematical Background of Classical Potential Theory	3
	Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions	14
	Infima of Families of Superharmonic Functions	35
	Potentials on Special Open Sets	45
	Polar Sets and Their Applications	57
	The Fundamental Convergence Theorem and the Reduction Operation	70
	Green Functions	85
	The Dirichlet Problem for Relative Harmonic Functions	98
	Lattices and Related Classes of Functions	141
	The Sweeping Operation	155
	The Fine Topology	166
	The Martin Boundary	195
	Classical Energy and Capacity	226
	One-Dimensional Potential Theory	256
	Parabolic Potential Theory: Basic Facts	262
	Subparabolic, Superparabolic, and Parabolic Functions on a Slab	285
	Parabolic Potential Theory (Continued)	295
	The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets	329
	The Martin Boundary in the Parabolic Context	363
	Probabilistic Counterpart of Part I
	Fundamental Concepts of Probability	387
	Optional Times and Associated Concepts	413
	Elements of Martingale Theory	432
	Basic Properties of Continuous Parameter Supermartingales	463
	Lattices and Related Classes of Stochastic Processes	520
	Markov Processes	539
	Brownian Motion	570
	The Ito Integral	599
	Brownian Motion and Martingale Theory	627
	Conditional Brownian Motion	668
	Lattices in Classical Potential Theory and Martingale Theory	705
	Brownian Motion and the PWB Method	719
	Brownian Motion on the Martin Space	727
	App. I: Analytic Sets	741
App. II	Capacity Theory	750
App. III	Lattice Theory	758
App. IV	Lattice Theoretic Concepts in Measure Theory	767
App. V	Uniform Integrability	779
App. VI	Kernels and Transition Functions	781
App. VII	Integral Limit Theorems	785
App. VIII	Lower Semicontinuous Functions	791
	Historical Notes	793
	Bibliography	819
	Notation Index	827
	Index	829