Author | Doob, J. L. author |
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Title | Classical Potential Theory and Its Probabilistic Counterpart [electronic resource] / by J. L. Doob |
Imprint | New York, NY : Springer New York, 1984 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5208-5 |
Descript | XXVI, 847 p. online resource |
1 Classical and Parabolic Potential Theory -- 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 -- 2 Probabilistic Counterpart of Part 1 -- 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 Itรด Integral -- IX Brownian Motion and Martingale Theory -- X Conditional Brownian Motion -- 3 -- 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 -- II Appendix -- Capacity Theory -- 1. Choquet Capacities -- 2. Sierpinski Lemma -- 3. Choquet Capacity Theorem -- 4. Lusinโs 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 -- III Appendix -- 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 -- IV Appendix -- Lattice Theoretic Concepts in Measure Theory -- 1. Lattices of Set Algebras -- 2. Measurable Spaces and Measurable Functions -- 3. Composition of Functions -- 4. The Measure Lattice of a Measurable Space -- 5. The o 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 -- V Appendix -- Uniform Integrability -- VI Appendix -- Kernels and Transition Functions -- 1. Kernels -- 2. Universally Measurable Extension of a Kernel -- 3. Transition Functions -- VII Appendix -- 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 -- VIII Appendix -- Lower Semicontinuous Functions -- The Lower Semicontinuous Smoothing of a Function -- Suprema of Families of Lower Semicontinuous Functions -- Choquet Topological Lemma -- Historical Notes -- 1 -- 2 -- 3 -- Appendixes -- Notation Index