Classical Potential Theory and Its Probabilistic Counterpart

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Main Author: Doob, Joseph L.
Corporate Author: SpringerLink (Online service)
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 One-Dimensional 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 One-Dimensional 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