Classical Potential Theory and Its Probabilistic Counterpart

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 Supermartingale.- 18. The ? Operator.- 19. The Natural Order Decomposition Theorem for Supermartingales.- 20. The Operators LM and GM.- 21. Supermartingale Potentials and the Riesz Decomposition.- 22. Potential Theory Reductions in a Discrete Parameter Probability Context.- 23. Application to the Crossing Inequalities.- IV Basic Properties of Continuous Parameter Supermartingales.- 1. Continuity Properties.- 2. Optional Sampling of Uniformly Integrable Continuous Parameter Martingales.- 3. Optional Sampling and Convergence of Continuous Parameter Supermartingales.- 4. Increasing Sequences of Supermartingales.- 5. Probability Version of the Fundamental Convergence Theorem of Potential Theory.- 6. Quasi-Bounded Positive Supermartingales; Generation of Supermartingale Potentials by Increasing Processes.- 7. Natural versus Predictable Increasing Processes (I=?+ or ?+).- 8. Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case.- 9. An Inequality for Predictable Increasing Processes.- 10. Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets.- 11. Generation of Supermartingale Potentials by Increasing Processes in the Continuous Parameter Case: The Meyer Decomposition.- 12. Meyer Decomposition of a Submartingale.- 13. Role of the Measure Associated with a Supermartingale; The Supermartingale Domination Principle.- 14. The Operators ?, LM, and GM in the Continuous Parameter Context.- 15. Potential Theory on ?+ ×?.- 16. The Fine Topology of ?+ ×?.- 17. Potential Theory Reductions in a Continuous Parameter Probability Context.- 18. Reduction Properties.- 19. Proofs of the Reduction Properties in Section 18.- 20. Evaluation of Reductions.- 21. The Energy of a Supermartingale Potential.- 22. The Subtraction of a Supermartingale Discontinuity.- 23. Supermartingale Decompositions and Discontinuities.- V Lattices and Related Classes of Stochastic Processes.- 1. Conventions; The Essential Order.- 2. LMx() when {x(), ?()} Is a Submartingale.- 3. Uniformly Integrable Positive Submartingales.- 4. Lp Bounded Stochastic Processes (p ? 1).- 5. The Lattices $$ ('{{{\mathbf{S}}}^{\pm }},),('{{{\mathbf{S}}}^{ + }},),({{{\mathbf{S}}}^{\pm }},),({{{\mathbf{S}}}^{ + }},) $$.- 6. The Vector Lattices $$ ('{\mathbf{S}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$ and $$ ({\mathbf{S}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$.- 7. The Vector Lattices $$ ('{{{\mathbf{S}}}_{m}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$ and $$ ({{{\mathbf{S}}}_{m}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$.- 8. The Vector Lattices $$ ({{{\mathbf{S}}}_{m}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$ and $$ ({{{\mathbf{S}}}_{p}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$.- 9. The Vector Lattices $$ ('{{{\mathbf{S}}}_{{qb}}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$ and $$ ({{{\mathbf{S}}}_{{qb}}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$.- 10. The Vector Lattices $$ ('{{{\mathbf{S}}}_{s}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$ and $$ ({{{\mathbf{S}}}_{s}},{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }}) $$.- 11. The Orthogonal Decompositions ´Sm = ´mqb+´Sms and Sm = Smqb + Sms.- 12. Local Martingales and Singular Supermartingale Potentials in (S, $$ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec }} $$).- 13. Quasimartingales (Continuous Parameter Context).- VI Markov Processes.- 1. The Markov Property.- 2. Choice of Filtration.- 3. Integral Parameter Markov Processes with Stationary Transition Probabilities.- 4. Application of Martingale Theory to Discrete Parameter Markov Processes.- 5. Continuous Parameter Markov Processes with Stationary Transition Probabilities.- 6. Specialization to Right Continuous Processes.- 7. Continuous Parameter Markov Processes: Lifetimes and Trap Points.- 8. Right Continuity of Markov Process Filtrations; A Zero-One (0-1) Law.- 9. Strong Markov Property.- 10. Probabilistic Potential Theory; Excessive Functions.- 11. Excessive Functions and Supermartingales.- 12. Excessive Functions and the Hitting Times of Analytic Sets (Notation and Hypotheses of Section 11).- 13. Conditioned Markov Processes.- 14. Tied Down Markov Processes.- 15. Killed Markov Processes.- VII Brownian Motion.- 1. Processes with Independent Increments and State Space ?N.- 2. Brownian Motion.- 3. Continuity of Brownian Paths.- 4. Brownian Motion Filtrations.- 5. Elementary Properties of the Brownian Transition Density and Brownian Motion.- 6. The Zero-One Law for Brownian Motion.- 7. Tied Down Brownian Motion.- 8. Andre Reflection Principle.- 9. Brownian Motion in an Open Set (N ? 1).- 10. Space-Time Brownian Motion in an Open Set.- 11. Brownian Motion in an Interval.- 12. Probabilistic Evaluation of Parabolic Measure for an Interval.- 13. Probabilistic Significance of the Heat Equation and Its Dual.- VIII The Itô Integral.- 1. Notation.- 2. The Size of ?0.- 3. Properties of the Itô Integral.- 4. The Stochastic Integral for an Integrand Process in ?0.- 5. The Stochastic Integral for an Integrand Process in ?.- 6. Proofs of the Properties in Section 3.- 7. Extension to Vector-Valued and Complex-Valued Integrands.- 8. Martingales Relative to Brownian Motion Filtrations.- 9. A Change of Variables.- 10. The Role of Brownian Motion Increments.- 11. (N = 1) Computation of the Ito Integral by Riemann-Stieltjes Sums.- 12. Itô's Lemma.- 13. The Composition of the Basic Functions of Potential Theory with Brownian Motion.- 14. The Composition of an Analytic Function with Brownian Motion.- IX Brownian Motion and Martingale Theory.- 1. Elementary Martingale Applications.- 2. Coparabolic Polynomials and Martingale Theory.- 3. Superharmonic and Harmonic Functions on ?N and Supermartingales and Martingales.- 4. Hitting of an F? Set.- 5. The Hitting of a Set by Brownian Motion.- 6. Superharmonic Functions, Excessive for Brownian Motion.- 7. Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion; A Probabilistic Fatou Boundary Limit Theorem.- 8. Excessive and Invariant Functions for Brownian Motion.- 9. Application to Hitting Probabilities and to Parabolicity of Transition Densities.- 10. (N = 2). The Hitting of Nonpolar Sets by Brownian Motion.- 11. Continuity of the Composition of a Function with Brownian Motion.- 12. Continuity of Superharmonic Functions on Brownian Motion.- 13. Preliminary Probabilistic Solution of the Classical Dirichlet Problem.- 14. Probabilistic Evaluation of Reductions.- 15. Probabilistic Description of the Fine Topology.- 16. ?-Excessive Functions for Brownian Motion and Their Composition with q Brownian Motions.- 17. Brownian Motion Transition Functions as Green Functions; The Corresponding Backward and Forward Parabolic Equations.- 18. Excessive Measures for Brownian Motion.- 19. Nearly Borel Sets for Brownian Motion.- 20. Brownian Motion into a Set from an Irregular Boundary Point.- X Conditional Brownian Motion.- 1. Definition.- 2. h-Brownian Motion in Terms of Brownian Motion.- 3. Contexts for (2.1).- 4. Asymptotic Character of h-Brownian Paths at Their Lifetimes.- 5. h-Brownian Motion from an Infinity of h.- 6. Brownian Motion under Time Reversal.- 7. Preliminary Probabilistic Solution of the Dirichlet Problem for h-Harmonic Functions; h-Brownian Motion Hitting Probabilities and the Corresponding Generalized Reductions.- 8. Probabilistic Boundary Limit and Internal Limit Theorems for Ratios of Strictly Positive Superharmonic Functions.- 9. Conditional Brownian Motion in a Ball.- 10. Conditional Brownian Motion Last Hitting Distributions; The Capacitary Distribution of a Set in Terms of a Last Hitting Distribution.- 11. The Tail ? Algebra of a Conditional Brownian Motion.- 12. Conditional Space-Time Brownian Motion.- 13. [Space-Time] Brownian Motion in $$ \left[ {{{{\dot{\mathbb{R}}}}^{N}}} \right]{{\mathbb{R}}^{N}} $$ with Parameter Set ?.- 3.- I Lattices in Classical Potential Theory and Martingale Theory.- 1. Correspondence between Classical Potential Theory and Martingale Theory.- 2. Relations between Decomposition Components of S in Potential Theory and Martingale Theory.- 3. The Classes Lp and D.- 4. PWB-Related Conditions on h-Harmonic Functions and on Martingales.- 5. Class D Property versus Quasi-Boundedness.- 6. A Condition for Quasi-Boundedness.- 7. Singularity of an Element of $$ {\mathbf{S}}_{m}^{ + } $$.- 8. The Singular Component of an Element of S+.- 9. The Class Spqb.- 10. The Class Sps.- 11. Lattice Theoretic Analysis of the Composition of an h-Superharmonic Function with an h-Brownian Motion.- 12. A Decomposition of $$ {\mathbf{S}}_{{ms}}^{ + } $$(Potential Theory Context).- 13. Continuation of Section 11.- II Brownian Motion and the PWB Method.- 1. Context of the Problem.- 2. Probabilistic Analysis of the PWB Method.- 3. PWBh Examples.- 4. Tail ? Algebras in the PWBh Context.- III Brownian Motion on the Martin Space.- 1. The Structure of Brownian Motion on the Martin Space.- 2. Brownian Motions from Martin Boundary Points (Notation of Section 1).- 3. The Zero-One Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the Minimal-Fine Topology (Notation of Section 1).- 4. The Probabilistic Fatou Theorem on the Martin Space.- 5. Probabilistic Approach to Theorem 1.XI.4(c) and Its Boundary Counterparts.- 6. Martin Representation of Harmonic Functions in the Parabolic Context.- 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. 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.- 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.