Classical Potential Theory and Its Probabilistic Counterpart
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 pap...
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Corporate Author:  
Format:  eBook 
Language:  English 
Published: 
Berlin, Heidelberg
Springer Berlin Heidelberg
2001, 2001

Edition:  1st ed. 2001 
Series:  Classics in Mathematics

Subjects:  
Online Access:  
Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 I Introduction to the Mathematical Background of Classical Potential Theory
 II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
 III Infima of Families of Superharmonic Functions
 IV Potentials on Special Open Sets
 V Polar Sets and Their Applications
 VI The Fundamental Convergence Theorem and the Reduction Operation
 VII Green Functions
 VIII The Dirichlet Problem for Relative Harmonic Functions
 IX Lattices and Related Classes of Functions
 X The Sweeping Operation
 XI The Fine Topology
 XII The Martin Boundary
 XIII Classical Energy and Capacity
 XIV OneDimensional Potential Theory
 XV Parabolic Potential Theory: Basic Facts
 XVI Subparabolic, Superparabolic, and Parabolic Functions on a Slab
 XVII Parabolic Potential Theory (Continued)
 XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
 XIX The Martin Boundary in the Parabolic Context
 I Fundamental Concepts of Probability
 II Optional Times and Associated Concepts
 III Elements of Martingale Theory
 IV Basic Properties of Continuous Parameter Supermartingales
 V Lattices and Related Classes of Stochastic Processes
 VI Markov Processes
 VII Brownian Motion
 VIII The Itô Integral
 IX Brownian Motion and Martingale Theory
 X Conditional Brownian Motion
 I Lattices in Classical Potential Theory and Martingale Theory
 II Brownian Motion and the PWB Method
 III Brownian Motion on the Martin Space
 Appendixes
 Appendix I
 Analytic Sets
 1. Pavings and Algebras of Sets
 2. Suslin Schemes
 3. Sets Analytic over a Product Paving
 4. Analytic Extensions versus ? Algebra Extensions of Pavings
 7. Projections of Sets in Product Pavings
 8. Extension of a Measurability Concept to the Analytic Operation Context
 10. Polish Spaces
 11. The Baire Null Space
 12. Analytic Sets
 13. Analytic Subsets of Polish Spaces
 Appendix II
 Capacity Theory
 1. Choquet Capacities
 2. Sierpinski Lemma
 3. Choquet Capacity Theorem
 4. Lusin’s Theorem
 5. A Fundamental Example of a Choquet Capacity
 6. Strongly Subadditive Set Functions
 7. Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function
 8. Topological Precapacities
 9. Universally Measurable Sets
 Appendix III
 Lattice Theory
 1. Introduction
 2. Lattice Definitions
 3. Cones
 4. The Specific Order Generated by a Cone
 5. Vector Lattices
 6. Decomposition Property of a Vector Lattice
 7. Orthogonality in a Vector Lattice
 8. Bands in a Vector Lattice
 9. Projections on Bands
 10. The Orthogonal Complement of a Set
 11. The Band Generated by a Single Element
 12. Order Convergence
 13. Order Convergence on a Linearly Ordered Set
 Appendix IV
 Lattice Theoretic Concepts in Measure Theory
 1. Lattices of Set Algebras
 2. Measurable Spaces and Measurable Functions
 3. Composition of Functions
 3. Choquet Topological Lemma
 Historical Notes
 1
 2
 3
 Appendixes
 Notation Index
 4. The Measure Lattice of a Measurable Space
 5. The ? Finite Measure Lattice of a Measurable Space (Notation of Section 4)
 6. The Hahn and Jordan Decompositions
 8. Absolute Continuity and Singularity
 9. Lattices of Measurable Functions on a Measure Space
 10.Order Convergence of Families of Measurable Functions
 11. Measures on Polish Spaces
 12. Derivates of Measures
 Appendix V
 Uniform Integrability
 Appendix VI
 Kernels and Transition Functions
 1. Kernels
 2. Universally Measurable Extension of a Kernel
 3. Transition Functions
 Appendix VII
 Integral Limit Theorems
 1. An Elementary Limit Theorem
 2. Ratio Integral Limit Theorems
 3. A OneDimensional Ratio Integral Limit Theorem
 4. A Ratio Integral Limit Theorem Involving Convex Variational Derivates
 Appendix VIII
 Lower Semicontinuous Functions
 1. The Lower Semicontinuous Smoothing of a Function
 2. Suprema of Families of Lower Semicontinuous Functions