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 papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. . . . The purpose which the author explains in his introduction, i. e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner".
M. Brelot in Metrika (1986)
Inhaltsverzeichnis
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. - 1. - 2. - 3. - Appendixes. - Notation Index.
