Classical Potential Theory and Its Probabilistic Counterpart: Advanced Problems: 262 - Brossura

Doob, J. L.

Brossura

ISBN 10: 1461297389 ISBN 13: 9781461297383

Casa editrice: Springer Verlag, 2011

Riassunto

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 withjaun­ 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 super­ 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.

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Contenuti

1 Classical and Parabolic Potential Theory.- I Introduction to the Mathematical Background of Classical Potential Theory.- 1. The Context of Green’s Identity.- 2. Function Averages.- 3. Harmonic Functions.- 4. Maximum-Minimum Theorem for Harmonic Functions.- 5. The Fundamental Kernel for ?N and Its Potentials.- 6. Gauss Integral Theorem.- 7. The Smoothness of Potentials; The Poisson Equation.- 8. Harmonic Measure and the Riesz Decomposition.- II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions.- 1. The Green Function of a Ball; The Poisson Integral.- 2. Harnack’s Inequality.- 3. Convergence of Directed Sets of Harmonic Functions.- 4. Harmonic, Subharmonic, and Superharmonic Functions.- 5. Minimum Theorem for Superharmonic Functions.- 6. Application of the Operation ?B.- 7. Characterization of Superharmonic Functions in Terms of Harmonic Functions.- 8. Differentiate Superharmonic Functions.- 9. Application of Jensen’s Inequality.- 10. Superharmonic Functions on an Annulus.- 11. Examples.- 12. The Kelvin Transformation (N ? 2).- 13. Greenian Sets.- 14. The L1(?B-) and D(?B-) Classes of Harmonic Functions on a Ball B; The Riesz-Herglotz Theorem.- 15. The Fatou Boundary Limit Theorem.- 16. Minimal Harmonic Functions.- III Infima of Families of Superharmonic Functions.- 1. Least Superharmonic Majorant (LM) and Greatest Subharmonic Minorant (GM).- 2. Generalization of Theorem 1.- 3. Fundamental Convergence Theorem (Preliminary Version).- 4. The Reduction Operation.- 5. Reduction Properties.- 6. A Smallness Property of Reductions on Compact Sets.- 7. The Natural (Pointwise) Order Decomposition for Positive Superharmonic Functions.- IV Potentials on Special Open Sets.- 1. Special Open Sets, and Potentials on Them.- 2. Examples.- 3. A Fundamental Smallness Property of Potentials.- 4. Increasing Sequences of Potentials.- 5. Smoothing of a Potential.- 6. Uniqueness of the Measure Determining a Potential.- 7. Riesz Measure Associated with a Superharmonic Function.- 8. Riesz Decomposition Theorem.- 9. Counterpart for Superharmonic Functions on ?2 of the Riesz Decomposition.- 10. An Approximation Theorem.- V Polar Sets and Their Applications.- 1. Definition.- 2. Superharmonic Functions Associated with a Polar Set.- 3. Countable Unions of Polar Sets.- 4. Properties of Polar Sets.- 5. Extension of a Superharmonic Function.- 6. Greenian Sets in ?2 as the Complements of Nonpolar Sets.- 7. Superharmonic Function Minimum Theorem (Extension of Theorem II.5).- 8. Evans-Vasilesco Theorem.- 9. Approximation of a Potential by Continuous Potentials.- 10. The Domination Principle.- 11. The Infinity Set of a Potential and the Riesz Measure.- VI The Fundamental Convergence Theorem and the Reduction Operation.- 1. The Fundamental Convergence Theorem.- 2. Inner Polar versus Polar Sets.- 3. Properties of the Reduction Operation.- 4. Proofs of the Reduction Properties.- 5. Reductions and Capacities.- VII Green Functions.- 1. Definition of the Green Function GD.- 2. Extremal Property of GD.- 3. Boundedness Properties of GD.- 4. Further Properties of GD.- 5. The Potential GD? of a Measure ?.- 6. Increasing Sequences of Open Sets and the Corresponding Green Function Sequences.- 7. The Existence of GD versus the Greenian Character of D.- 8. From Special to Greenian Sets.- 9. Approximation Lemma.- 10. The Function $$ {{G}_{D}}{{( \bullet ,\zeta )}_{{\left| {D - \left\{ \zeta \right\}} \right.}}} $$ as a Minimal Harmonic Function.- VIII The Dirichlet Problem for Relative Harmonic Functions.- 1. Relative Harmonic, Superharmonic, and Subharmonic Functions.- 2. The PWB Method.- 3. Examples.- 4. Continuous Boundary Functions on the Euclidean Boundary (h ? 1).- 5. h-Harmonic Measure Null Sets.- 6. Properties of PWBh Solutions.- 7. Proofs for Section 6.- 8. h-Harmonic Measure.- 9. h-Resolutive Boundaries.- 10. Relations between Reductions and Dirichlet Solutions.- 11. Generalization of the Operator $$ \tau _{B}^{h} $$ and Application to GMh.- 12. Barriers.- 13. h-Barriers and Boundary Point h-Regularity.- 14. Barriers and Euclidean Boundary Point Regularity.- 15. The Geometrical Significance of Regularity (Euclidean Boundary, h ? 1).- 16. Continuation of Section 13.- 17. h-Harmonic Measure $$ \mu _{D}^{h} $$ as a Function of D.- 18. The Extension $$ G_{D}^{ = } $$ of GD and the Harmonic Average $$ {{\mu }_{D}}(\xi ,G_{B}^{ = }(\eta , \bullet )) $$ When D?B.- 19. Modification of Section 18 for D = ?2.- 20. Interpretation of ?D as a Green Function with Pole ? (N = 2).- 21. Variant of the Operator ?B.- IX Lattices and Related Classes of Functions.- 1. Introduction.- 2. $$ {\text{LM}}_{D}^{h}u $$ for an h-Subharmonic Function u.- 3. The Class D$$ (\mu _{{D - }}^{h}) $$.- 4. The Class Lp$$ (\mu _{{D - }}^{h})(p1) $$.- 5. The Lattices $$ ({{{\mathbf{S}}}^{\pm }},) $$ and (S+, ?).- 6. The Vector Lattice (S, $$ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }} $$).- 7. The Vector Lattice Sm.- 8. The Vector Lattice Sp.- 9. The Vector Lattice Sqb.- 10. The Vector Lattice Ss.- 11. A Refinement of the Riesz Decomposition.- 12. Lattices of h-Harmonic Functions on a Ball.- X The Sweeping Operation.- 1. Sweeping Context and Terminology.- 2. Relation between Harmonic Measure and the Sweeping Kernel.- 3. Sweeping Symmetry Theorem.- 4. Kernel Property of $$ \delta _{D}^{A} $$.- 5. Swept Measures and Functions.- 6. Some Properties of$$ \delta _{D}^{A} $$.- 7. Poles of a Positive Harmonic Function.- 8. Relative Harmonic Measure on a Polar Set.- XI The Fine Topology.- 1. Definitions and Basic Properties.- 2. A Thinness Criterion.- 3. Conditions That ??Af.- 4. An Internal Limit Theorem.- 5. Extension of the Fine Topology to $$ {{\mathbb{R}}^{N}} \cup \left\{ \infty \right\} $$.- 6. The Fine Topology Derived Set of a Subset of ?N.- 7. Application to the Fundamental Convergence Theorem and to Reductions.- 8. Fine Topology Limits and Euclidean Topology Limits.- 9. Fine Topology Limits and Euclidean Topology Limits (Continued).- 10. Identification of Af in Terms of a Special Function u#.- 11. Quasi-Lindelöf Property.- 12. Regularity in Terms of the Fine Topology.- 13. The Euclidean Boundary Set of Thinness of a Greenian Set.- 14. The Support of a Swept Measure.- 15. Characterization of ???A.- 16. A Special Reduction.- 17. The Fine Interior of a Set of Constancy of a Superharmonic Function.- 18. The Support of a Swept Measure (Continuation of Section 14).- 19. Superharmonic Functions on Fine-Open Sets.- 20. A Generalized Reduction.- 21. Limits of Superharmonic Functions at Irregular Boundary Points of Their Domains.- 22. The Limit Harmonic Measure f?D.- 23. Extension of the Domination Principle.- XII The Martin Boundary.- 1. Motivation.- 2. The Martin Functions.- 3. The Martin Space.- 4. Preliminary Representations of Positive Harmonic Functions and Their Reductions.- 5. Minimal Harmonic Functions and Their Poles.- 6. Extension of Lemma 4.- 7. The Set of Nonminimal Martin Boundary Points.- 8. Reductions on the Set of Minimal Martin Boundary Points.- 9. The Martin Representation.- 10. Resolutivity of the Martin Boundary.- 11. Minimal Thinness at a Martin Boundary Point.- 12. The Minimal-Fine Topology.- 13. First Martin Boundary Counterpart of Theorem XI.4(c) and (d).- 14. Second Martin Boundary Counterpart of Theorem XI.4(c).- 15. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point.- 16. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point (Continued).- 17. Minimal-Fine Martin Boundary Limit Functions.- 18. The Fine Boundary Function of a Potential.- 19. The Fatou Boundary Limit Theorem for the Martin Space.- 20. Classical versus Minimal-Fine Topology Boundary Limit Theorems for Relative Superharmonic Functions on a Ball in ?N.- 21. Nontangential and Minimal-Fine Limits at a Half-space Boundary.- 22. Normal Boundary Limits for a Half-space.- 23. Boundary Limit Function (Minimal-Fine and Normal) of a Potential on a Half-space.- XIII Classical Energy and Capacity.- 1. Physical Context.- 2. Measures and Their Energies.- 3. Charges and Their Energies.- 4. Inequalities between Potentials, and the Corresponding Energy Inequalities.- 5. The Function D?GD?.- 6. Classical Evaluation of Energy; Hilbert Space Methods.- 7. The Energy Functional (Relative to an Arbitrary Greenian Subset D of ?N).- 8. Alternative Proofs of Theorem 7(b+).- 9. Sharpening of Lemma 4.- 10. The Classical Capacity Function.- 11. Inner and Outer Capacities (Notation of Section 10).- 12. Extremal Property Characterizations of Equilibrium Potentials (Notation of Section 10).- 13. Expressions for C(A).- 14. The Gauss Minimum Problems and Their Relation to Reductions.- 15. Dependence of C* on D.- 16. Energy Relative to ?2.- 17. The Wiener Thinness Criterion.- 18. The Robin Constant and Equilibrium Measures Relative to ?2 (N = 2).- XIV One-Dimensional Potential Theory.- 1. Introduction.- 2. Harmonic, Superharmonic, and Subharmonic Functions.- 3. Convergence Theorems.- 4. Smoothness Properties of Superharmonic and Subharmonic Functions.- 5. The Dirichlet Problem (Euclidean Boundary).- 6. Green Functions.- 7. Potentials of Measures.- 8. Identification of the Measure Defining a Potential.- 9. Riesz Decomposition.- 10. The Martin Boundary.- XV Parabolic Potential Theory: Basic Facts.- 1. Conventions.- 2. The Parabolic and Coparabolic Operators.- 3. Coparabolic Polynomials.- 4. The Parabolic Green Function of $$ {{\dot{\mathbb{R}}}^{N}} $$.- 5. Maximum-Minimum Parabolic Function Theorem.- 6. Application of Green’s Theorem.- 7. The Parabolic Green Function of a Smooth Domain; The Riesz Decomposition and Parabolic Measure (Formal Treatment).- 8. The Green Function of an Interval.- 9. Parabolic Measure for an Interval.- 10. Parabolic Averages.- 11. Harnack’s Theorems in the Parabolic Context.- 12. Superparabolic Functions.- 13. Superparabolic Function Minimum Theorem.- 14. The Operation $$ {{\dot{\tau }}_{{\dot{B}}}} $$ and the Defining Average Properties of Superparabolic Functions.- 15. Superparabolic and Parabolic Functions on a Cylinder.- 16. The Appell Transformation.- 17. Extensions of a Parabolic Function Defined on a Cylinder.- XVI Subparabolic, Superparabolic, and Parabolic Functions on a Slab.- 1. The Parabolic Poisson Integral for a Slab.- 2. A Generalized Superparabolic Function Inequality.- 3. A Criterion of a Subparabolic Function Supremum.- 4. A Boundary Limit Criterion for the Identically Vanishing of a Positive Parabolic Function.- 5. A Condition that a Positive Parabolic Function Be Representable by a Poisson Integral.- 6. The $$ {{{\mathbf{L}}}^{1}}({{\dot{\mu }}_{{\dot{B} - }}}) $$ and $$ {\mathbf{D}}({{\dot{\mu }}_{{\dot{B} - }}}) $$ Classes of Parabolic Functions on a Slab.- 7. The Parabolic Boundary Limit Theorem.- 8. Minimal Parabolic Functions on a Slab.- XVII Parabolic Potential Theory (Continued).- 1. Greatest Minorants and Least Majorants.- 2. The Parabolic Fundamental Convergence Theorem (Preliminary Version) and the Reduction Operation.- 3. The Parabolic Context Reduction Operations.- 4. The Parabolic Green Function.- 5. Potentials.- 6. The Smoothness of Potentials.- 7. Riesz Decomposition Theorem.- 8. Parabolic-Polar Sets.- 9. The Parabolic-Fine Topology.- 10. Semipolar Sets.- 11. Preliminary List of Reduction Properties.- 12. A Criterion of Parabolic Thinness.- 13. The Parabolic Fundamental Convergence Theorem.- 14. Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions.- 15. Applications of the Fundamental Convergence Theorem to the Parabolic-Fine Topology.- 16. Parabolic-Reduction Properties.- 17. Proofs of the Reduction Properties in Section 16.- 18. The Classical Context Green Function in Terms of the Parabolic Context Green Function (N ? 1).- 19. The Quasi-Lindelöf Property.- XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets.- 1. Relativization of the Parabolic Context; The PWB Method in this Context.- 2. $$ \dot{h} $$-Parabolic Measure.- 3. Parabolic Barriers.- 4. Relations between the Classical Dirichlet Problem and the Parabolic Context Dirichlet Problem.- 5. Classical Reductions in the Parabolic Context.- 6. Parabolic Regularity of Boundary Points.- 7. Parabolic Regularity in Terms of the Fine Topology.- 8. Sweeping in the Parabolic Context.- 9. The Extension $$ \dot{G}_{{\dot{D}}}^{ = } $$ of $$ {{\dot{G}}_{{\dot{D}}}} $$ and the Parabolic Average $$ {{\dot{\mu }}_{{\dot{D}}}}(\dot{\xi },\dot{G}_{{\dot{B}}}^{ = }( \bullet ,\dot{\eta })) $$ when $$ \dot{D} \subset \dot{B} $$.- 10. Conditions that $$ \dot{\xi } \in {{\dot{A}}^{{pf}}} $$.- 11. Parabolic-and Coparabolic-Polar Sets.- 12. Parabolic- and Coparabolic-Semipolar Sets.- 13. The Support of a Swept Measure.- 14. An Internal Limit Theorem; The Coparabolic-Fine Topology Smoothness of Superparabolic Functions.- 15. Application to a Version of the Parabolic Context Fatou Boundary Limit Theorem on a Slab.- 16. The Parabolic Context Domination Principle.- 17. Limits of Superparabolic Functions at Parabolic-Irregular Boundary Points of Their Domains.- 18. Martin Flat Point Set Pairs.- 19. Lattices and Related Classes of Functions in the Parabolic Context.- XIX The Martin Boundary in the Parabolic Context.- 1. Introduction.- 2. The Martin Functions of Martin Point Set and Measure Set Pairs.- 3. The Martin Space $$ {{\dot{D}}^{M}} $$.- 4. Preparatory Material for the Parabolic Context Martin Representation Theorem.- 5. Minimal Parabolic Functions and Their Poles.- 6. The Set of Nonminimal Martin Boundary Points.- 7. The Martin Representation in the Parabolic Context.- 8. Martin Boundary of a Slab $$ \dot{D} = {{\mathbb{R}}^{N}} \times \left] {0,\delta } \right[ $$ with 0 Context Fatou Boundary Limit Theorem on Martin Spaces.- 2 Probabilistic Counterpart of Part 1.- I Fundamental Concepts of Probability.- 1. Adapted Families of Functions on Measurable Spaces.- 2. Progressive Measurability.- 3. Random Variables.- 4. Conditional Expectations.- 5. Conditional Expectation Continuity Theorem.- 6. Fatou’s Lemma for Conditional Expectations.- 7. Dominated Convergence Theorem for Conditional Expectations.- 8. Stochastic Processes, “Evanescent,” “Indistinguishable,” “Standard Modification,” “Nearly”.- 9. The Hitting of Sets and Progressive Measurability.- 10. Canonical Processes and Finite-Dimensional Distributions.- 11. Choice of the Basic Probability Space.- 12. The Hitting of Sets by a Right Continuous Process.- 13. Measurability versus Progressive Measurability of Stochastic Processes.- 14. Predictable Families of Functions.- II Optional Times and Associated Concepts.- 1. The Context of Optional Times.- 2. Optional Time Properties (Continuous Parameter Context).- 3. Process Functions at Optional Times.- 4. Hitting and Entry Times.- 5. Application to Continuity Properties of Sample Functions.- 6. Continuation of Section 5.- 7. Predictable Optional Times.- 8. Section Theorems.- 9. The Graph of a Predictable Time and the Entry Time of a Predictable Set.- 10. Semipolar Subsets of ?+ × ?.- 11. The Classes D and Lp of Stochastic Processes.- 12. Decomposition of Optional Times; Accessible and Totally Inaccessible Optional Times.- III Elements of Martingale Theory.- 1. Definitions.- 2. Examples.- 3. Elementary Properties (Arbitrary Simply Ordered Parameter Set).- 4. The Parameter Set in Martingale Theory.- 5. Convergence of Supermartingale Families.- 6. Optional Sampling Theorem (Bounded Optional Times).- 7. Optional Sampling Theorem for Right Closed Processes.- 8. Optional Stopping.- 9. Maximal Inequalities.- 10. Conditional Maximal Inequalities.- 11. An LL Inequality for Submartingale Suprema.- 12. Crossings.- 13. Forward Convergence in the L1 Bounded Case.- 14. Convergence of a Uniformly Integrable Martingale.- 15. Forward Convergence of a Right Closable Supermartingale.- 16. Backward Convergence of a Martingale.- 17. Backward Convergence of a Super...

Dettagli dell?edizione

• Editore: Springer Verlag
• Data di pubblicazione: 2011
• Lingua: Inglese
• ISBN 10: 1461297389
• ISBN 13: 9781461297383
• Rilegatura: Copertina flessibile
• Numero di pagine: 847
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile