Author | Doob, J. L. author |
---|---|

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 |

SUMMARY

Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaunยญ diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of superยญ martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on

CONTENT

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. Lusinโ{128}{153}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 -- 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

Mathematics
Potential theory (Mathematics)
Probabilities
Mathematics
Potential Theory
Probability Theory and Stochastic Processes