Hjem

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.-
4. The Measure Lattice of a Measurable Space.-
5. The ? Finite Measure Lattice of a Measurable Space (Notation of Section 4).-
6. TheHahn 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.-
3. Choquet Topological Lemma.- Historical Notes.-
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2.-
3.- Appendixes.- Notation Index.