J. L. Doob

Classical Potential Theoryand Its ProbabilisticCounterpart

Springer-VerlagNew York Berlin Heidelberg Tokyo

Contents

Introduction xxiNotation and Conventions xxv

Part 1Classical and Parabolic Potential Theory

Chapter IIntroduction to the Mathematical Background of Classical PotentialTheory 3

1. The Context of Green's Identity 32. Function Averages 43. Harmonic Functions 44. Maximum-Minimum Theorem for Harmonic Functions 55. The Fundamental Kernel for RN and Its Potentials 66. Gauss Integral Theorem 77. The Smoothness of Potentials; The Poisson Equation 88. Harmonic Measure and the Riesz Decomposition 11

Chapter IIBasic Properties of Harmonic, Subharmonic, and SuperharmonicFunctions 14

1. The Green Function of a Ball; The Poisson Integral 142. Harnack's Inequality 163. Convergence of Directed Sets of Harmonic Functions 174. Harmonic, Subharmonic, and Superharmonic Functions 185. Minimum Theorem for Superharmonic Functions 206. Application of the Operation rB 207. Characterization of Superharmonic Functions in Terms of Harmonic

Functions 228. Differentiable Superharmonic Functions 239. Application of Jensen's Inequality 23

10. Superharmonic Functions on an Annulus 2411. Examples 2512. The Kelvin Transformation (N>2) 26

vi Contents

13. Greenian Sets 2714. The L'(//fl_) and D(fiB_) Classes of Harmonic Functions on a Ball B; The

Riesz-Herglotz Theorem 2715. The Fatou Boundary Limit Theorem 3116. Minimal Harmonic Functions 33

Chapter IIIInfima of Families of Superharmonic Functions 35

1. Least Superharmonic Majorant (LM) and Greatest SubharmonicMinorant (GM) 35

2. Generalization of Theorem 1 363. Fundamental Convergence Theorem (Preliminary Version) 374. The Reduction Operation 385. Reduction Properties 416. A Smallness Property of Reductions on Compact Sets 427. The Natural (Pointwise) Order Decomposition for Positive Superharmonic

Functions 43

Chapter IVPotentials on Special Open Sets 45

1. Special Open Sets, and Potentials on Them 452. Examples 473. A Fundamental Smallness Property of Potentials 484. Increasing Sequences of Potentials 495. Smoothing of a Potential 496. Uniqueness of the Measure Determining a Potential 507. Riesz Measure Associated with a Superharmonic Function 518. Riesz Decomposition Theorem L. 529. Counterpart for Superharmonic Functions on U2 of the Riesz

Decomposition 5310. An Approximation Theorem 55

Chapter VPolar Sets and Their Applications 57

1. Definition 572. Superharmonic Functions Associated with a Polar Set 583. Countable Unions of Polar Sets 594. Properties of Polar Sets 595. Extension of a Superharmonic Function 606. Greenian Sets in R2 as the Complements of Nonpolar Sets 637. Superharmonic Function Minimum Theorem (Extension of

Theorem II.5) 638. Evans-Vasilesco Theorem 649. Approximation of a Potential by Continuous Potentials 66

10. The Domination Principle 6711. The Infinity Set of a Potential and the Riesz Measure 68

Contents Vll

Chapter VIThe Fundamental Convergence Theorem and the ReductionOperation 70

1. The Fundamental Convergence Theorem 702. Inner Polar versus Polar Sets 713. Properties of the Reduction Operation 744. Proofs of the Reduction Properties 775. Reductions and Capacities 84

Chapter VIIGreen Functions 85

1. Definition of the Green Function GD 852. Extremal Property of GD 873. Boundedness Properties of GD 884. Further Properties of GD 905. The Potential GDn of a Measure fj. 926. Increasing Sequences of Open Sets and the Corresponding Green Function

Sequences 947. The Existence of GD versus the Greenian Character of D 948. From Special to Greenian Sets 959. Approximation Lemma 95

10. The Function GD (SO|D-(O a s a Minimal Harmonic Function 96

Chapter VIIIThe Dirichlet Problem for Relative Harmonic Functions 98

1. Relative Harmonic, Superharmonic, and Subharmonic Functions 982. The PWB Method 993. Examples 1044. Continuous Boundary Functions on the Euclidean Boundary (h = 1) . . . . 1065. ft-Harmonic Measure Null Sets 1086. Properties of PWB* Solutions 1107. Proofs for Section 6 I l l8. /i-Harmonic Measure 1149. /i-Resolutive Boundaries 118

10. Relations between Reductions and Dirichlet Solutions 12211. Generalization of the Operator zB and Application to GM' 12312. Barriers 12413. /i-Barriers and Boundary Point ^-Regularity 12614. Barriers and Euclidean Boundary Point Regularity 12715. The Geometrical Significance of Regularity (Euclidean Boundary, h = 1). 12816. Continuation of Section 13 13017. /i-Harmonic Measure nh

D as a Function of D 13118. The Extension G^ of GD and the Harmonic Average iiD(£,, GB(r\, •)) When

D <= B 13219. Modification of Section 18 for D = R2 13620. Interpretation of <j>D as a Green Function with Pole oo (N = 2) 13921. Variant of the Operator xB 140

viii Contents

Chapter IXLattices and Related Classes of Functions 141

1. Introduction 1412. LM"0 u for an A-Subharmonic Function u 1413. The Class D(/x{,_) '. 1424. The Class L"(/iJ,_) (p > 1) 1445. The Lattices (S*, <) and (S+,<<) 1456. The Vector Lattice (S, <) 1467. The Vector Lattice Sm 1488. The Vector Lattice Sp 1499. The Vector Lattice Sqb 150

10. The Vector Lattice Ss 15111. A Refinement of the Riesz Decomposition 15212. Lattices of /j-Harmonic Functions on a Ball 152

Chapter XThe Sweeping Operation 155

1. Sweeping Context and Terminology 1552. Relation between Harmonic Measure and the Sweeping Kernel 1573. Sweeping Symmetry Theorem 1584. Kernel Property of d£ 1585. Swept Measures and Functions 1606. Some Properties of 5% 1617. Poles of a Positive Harmonic Function 1638. Relative Harmonic Measure on a Polar Set 164

Chapter XIThe Fine Topology 166

1. Definitions and Basic Properties 1662. A Thinness Criterion 1683. Conditions That £eAf 1694. An Internal Limit Theorem 1715. Extension of the Fine Topology to UN u {oo} 1756. The Fine Topology Derived Set of a Subset of RN 1777. Application to the Fundamental Convergence Theorem and to Reductions. 1778. Fine Topology Limits and Euclidean Topology Limits : 1789. Fine Topology Limits and Euclidean Topology Limits (Continued) 179

10. Identification of Af in Terms of a Special Function u* 18011. Quasi-Lindelof Property 18012. Regularity in Terms of the Fine Topology 18113. The Euclidean Boundary Set of Thinness of a Greenian Set 18214. The Support of a Swept Measure 18315. Characterization of \n\A 18316. A Special Reduction 18417. The Fine Interior of a Set of Constancy of a Superharmonic Function . . . 18418. The Support of a Swept Measure (Continuation of Section 14) 18519. Superharmonic Functions on Fine-Open Sets 18720. A Generalized Reduction 187

Contents IX

21. Limits of Superharmonic Functions at Irregular Boundary Points of TheirDomains 190

22. The Limit Harmonic Measure ffiD 19123. Extension of the Domination Principle 194

Chapter XIIThe Mar t in Boundary 195

1. Motivation 1952. The Martin Functions 1963. The Martin Space 1974. Preliminary Representations of Positive Harmonic Functions and Their

Reductions 1995. Minimal Harmonic Functions and Their Poles 2006. Extension of Lemma 4 2017. The Set of Nonminimal Martin Boundary Points 2028. Reductions on the Set of Minimal Martin Boundary Points 2039. The Martin Representation 204

10. Resolutivity of the Martin Boundary 20711. Minimal Thinness at a Martin Boundary Point 20812. The Minimal-Fine Topology 21013. First Martin Boundary Counterpart of Theorem XI.4(c) and (d) 21314. Second Martin Boundary Counterpart of Theorem XI.4(c) 21315. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal

Martin Boundary Point 21516. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal

Martin Boundary Point (Continued) 21617. Minimal-Fine Martin Boundary Limit Functions 21618. The Fine Boundary Function of a Potential 21819. The Fatou Boundary Limit Theorem for the Martin Space 21920. Classical versus Minimal-Fine Topology Boundary Limit Theorems for

Relative Superharmonic Functions on a Ball in RN 22121. Nontangential and Minimal-Fine Limits at a Half-space Boundary 22222. Normal Boundary Limits for a Half-space 22323. Boundary Limit Function (Minimal-Fine and Normal) of a Potential on a

Half-space 225

Chapter XIIIClassical Energy and Capaci ty 226

1. Physical Context 2262. Measures and Their Energies 2273. Charges and Their Energies 2284. Inequalities between Potentials, and the Corresponding Energy

Inequalities 2295. The Function D\^>GD\i 2306. Classical Evaluation of Energy; Hilbert Space Methods 2317. The Energy Functional (Relative to an Arbitrary Greenian Subset D of

W) 2338. Alternative Proofs of Theorem l(b+) 235

X Contents

9. Sharpening of Lemma 4 23710. The Classical Capacity Function 23711. Inner and Outer Capacities (Notation of Section 10) 24012. Extremal Property Characterizations of Equilibrium Potentials (Notation

of Section 10) 24113. Expressions for C(A) 24314. The Gauss Minimum Problems and Their Relation to Reductions 24415. Dependence of C* on D 24716. Energy Relative to R2 24817. The Wiener Thinness Criterion 24918. The Robin Constant and Equilibrium Measures Relative to R2 (N = 2) . . 251

Chapter XIVOne-Dimensional Potential Theory 256

1. Introduction ' 2562. Harmonic, Superharmonic, and Subharmonic Functions 2563. Convergence Theorems .• 2564. Smoothness Properties of Superharmonic and Subharmonic Functions... 2575. The Dirichlet Problem (Euclidean Boundary) 2576. Green Functions 2587. Potentials of Measures .• 2598. Identification of the Measure Defining a Potential •. 2599. Riesz Decomposition 260

10. The Martin Boundary 261

Chapter XVParabolic Potential Theory: Basic Facts 262

1. Conventions 2622. The Parabolic and Coparabolic Operators 2633. Coparabolic Polynomials 2644. The Parabolic Green Function of RN '. 2665. Maximum-Minimum Parabolic Function Theorem 2676. Application of Green's Theorem 2697. The Parabolic Green Function of a Smooth Domain; The Riesz Decom-

position and Parabolic Measure (Formal Treatment) 2708. The Green Function of an Interval 2729. Parabolic Measure for an Interval 273

10. Parabolic Averages 27511. Harnack's Theorems in the Parabolic Context 27612. Superparabolic Functions 27713. Superparabolic Function Minimum Theorem 27914. The Operation iB and the Defining Average Properties of Superparabolic

Functions 28015. Superparabolic and Parabolic Functions on a Cylinder 28116. The Appell Transformation 28217. Extensions of a Parabolic Function Defined on a Cylinder 283

Contents XI

Chapter XVISubparabolic, Superparabolic, and Parabolic Functions on a Slab . . . 285

1. The Parabolic Poisson Integral for a Slab 2852. A Generalized Superparabolic Function Inequality 2873. A Criterion of a Subparabolic Function Supremum 2884. A Boundary Limit Criterion for the Identically Vanishing of a Positive

Parabolic Function 2885. A Condition that a Positive Parabolic Function Be Representable by a

Poisson Integral 2906. The L'(/iB_) and D(/ig_) Classes of Parabolic Functions on a Slab 2907. The Parabolic Boundary Limit Theorem 2928. Minimal Parabolic Functions on a Slab 293

Chapter XVIIParabolic Potential Theory (Continued) 295

1. Greatest Minorants and Least Majorants 2952. The Parabolic Fundamental Convergence Theorem (Preliminary Version)

and the Reduction Operation 2953. The Parabolic Context Reduction Operations 2964. The Parabolic Green Function 2985. Potentials 3006. The Smoothness of Potentials 3037. Riesz Decomposition Theorem 3058. Parabolic-Polar Sets 3059. The Parabolic-Fine Topology 308

10. Semipolar Sets 30911. Preliminary List of Reduction Properties 31012. A Criterion of Parabolic Thinness 31313. The Parabolic Fundamental Convergence Theorem 31414. Applications of the Fundamental Convergence Theorem to Reductions

and to Green Functions 31615. Applications of the Fundamental Convergence Theorem to the Parabolic-

Fine Topology 31716. Parabolic-Reduction Properties 31717. Proofs of the Reduction Properties in Section 16 32018. The Classical Context Green Function in Terms of the Parabolic Context

Green Function (N > \) 32619. The Quasi-Lindelof Property 328

Chapter XVIIIT h e P a r a b o l i c Di r i ch le t P r o b l e m , Sweep ing , a n d E x c e p t i o n a l Sets . . . 329

1. Relativization of the Parabolic Context; The PWB Method in thisContext 329

2. A-Parabolic Measure 3323. Parabolic Barriers 3334. Relations between the Classical Dirichlet Problem and the Parabolic

Context Dirichlet Problem 3345. Classical Reductions in the Parabolic Context 335

Xli Contents

6. Parabolic Regularity of Boundary Points 3377. Parabolic Regularity in Terms of the Fine Topology 3418. Sweeping in the Parabolic Context 3419. The Extension G^ of G^ and the Parabolic Average /io(£> GJ (•, >/)) when

D <= B 34310. Conditions that £eArf 34511. Parabolic- and Coparabolic-Polar Sets 34712. Parabolic- and Coparabolic-Semipolar Sets 34813. The Support of a Swept Measure 35014. An Internal Limit Theorem; The Coparabolic-Fine Topology Smoothness

of Superparabolic Functions 35115. Application to a Version of the Parabolic Context Fatou Boundary Limit

Theorem on a Slab 35716. The Parabolic Context Domination Principle 35817. Limits of Superparabolic Functions at Parabolic-Irregular Boundary

Points of Their Domains 35818. Martin Flat Point Set Pairs 36119. Lattices and Related Classes of Functions in the Parabolic Context 361

Chapter XIXThe Martin Boundary in the Parabolic Context 363

1. Introduction 3632. The Martin Functions of Martin Point Set and Measure Set Pairs 3643. The Martin Space DM 3664. Preparatory Material for the Parabolic Context Martin Representation

Theorem 3675. Minimal Parabolic Functions and Their Poles 3696. The Set of Nonminimal Martin Boundary Points 3707. The Martin Representation in the Parabolic Context 3718. Martin Boundary of a Slab D = UN x ]0, <5[ with 0 < <5 < + oo 3719. Martin Boundaries for the Lower Half-space of RN and for RN 374

10. The Martin Boundary of D = ]0, + oo[ x ]-oo,<5[ 37511. PWB* Solutions on DM 37712. The Minimal-Fine Topology in the Parabolic Context 37713. Boundary Counterpart of Theorem XVIII. 14(f) 37914. The Vanishing of Potentials on 8MD 38115. The Parabolic Context Fatou Boundary Limit Theorem on Martin Spaces 381

Part 2Probabilistic Counterpart of Part 1

Chapter IFundamental Concepts of Probability 387

1. Adapted Families of Functions on Measurable Spaces 3872. Progressive Measurability 3883. Random Variables 390

Contents Xlll

4. Conditional Expectations 3915. Conditional Expectation Continuity Theorem 3936. Fatou's Lemma for Conditional Expectations 3967. Dominated Convergence Theorem for Conditional Expectations 3978. Stochastic Processes, "Evanescent," "Indistinguishable," "Standard Modi-

fication," "Nearly" , 3989. The Hitting of Sets and Progressive Measurability 401

10. Canonical Processes and Finite-Dimensional Distributions 40211. Choice of the Basic Probability Space 40412. The Hitting of Sets by a Right Continuous Process 40513. Measurability versus Progressive Measurability of Stochastic Processes . . 40714. Predictable Families of Functions 410

Chapter IIOptional Times and Associated Concepts 413

1. The Context of Optional Times 4132. Optional Time Properties (Continuous Parameter Context) 4153. Process Functions at Optional Times 4174. Hitting and Entry Times 4195. Application to Continuity Properties of Sample Functions 4216. Continuation of Section 5 4237. Predictable Optional Times 4238. Section Theorems 4259. The Graph of a Predictable Time and the Entry Time of a Predictable

Set 42610. Semipolar Subsets of U+ x il 42711. The Classes D and Lp of Stochastic Processes 42812. Decomposition of Optional Times; Accessible and Totally Inaccessible

Optional Times 429

Chapter IIIElements of Martingale Theory 432

1. Definitions 4322. Examples 4333. Elementary Properties (Arbitrary Simply Ordered Parameter Set) 4354. The Parameter Set in Martingale Theory 4375. Convergence of Supermartingale Families 4376. Optional Sampling Theorem (Bounded Optional Times) 4387. Optional Sampling Theorem for Right Closed Processes 4408. Optional Stopping 4429. Maximal Inequalities 442

10. Conditional Maximal Inequalities 44411. An W Inequality for Submartingale Suprema 44412. Crossings 44513. Forward Convergence in the L1 Bounded Case 45014. Convergence of a Uniformly Integrable Martingale 45115. Forward Convergence of a Right Closable Supermartingale 45316. Backward Convergence of a Martingale 454

xiv Contents

17. Backward Convergence of a Supermartingale 45518. The T Operator 45519. The Natural Order Decomposition Theorem for Supermartingales 45720. The Operators LM and GM -. 45821. Supermartingale Potentials and the Riesz Decomposition 45922. Potential Theory Reductions in a Discrete Parameter Probability

Context 45923. Application to the Crossing Inequalities 461

Chapter IVBasic Properties of Continuous Parameter Supermartingales 463

1. Continuity Properties 4632. Optional Sampling of Uniformly Integrable Continuous Parameter

Martingales 4683. Optional Sampling and Convergence of Continuous Parameter

Supermartingales 4704. Increasing Sequences of Supermartingales 4735. Probability Version of the Fundamental Convergence Theorem of Potential

Theory 4766. Quasi-Bounded Positive Supermartingales; Generation of Supermartingale

Potentials by Increasing Processes 4807. Natural versus Predictable Increasing Processes (/ = Z+ or IR+) 4838. Generation of Supermartingale Potentials by Increasing Processes in the

Discrete Parameter Case 4889. An Inequality for Predictable Increasing Processes 489

10. Generation of Supermartingale Potentials by Increasing Processes forArbitrary Parameter Sets 490

11. Generation of Supermartingale Potentials by Increasing Processes in theContinuous Parameter Case: The Meyer Decomposition 493

12. Meyer Decomposition of a Submartingale 49513. Role of the Measure Associated with a Supermartingale;

The Supermartingale Domination Principle 49614. The Operators T, LM, and GM in the Continuous Parameter Context. . . . 50015. Potential Theory on IR+ x Q 50116. The Fine Topology of R+ x fi 50217. Potential Theory Reductions in a Continuous Parameter Probability

Context 50418. Reduction Properties 50519. Proofs of the Reduction Properties in Section 18 50920. Evaluation of Reductions 51321. The Energy of a Supermartingale Potential 51522. The Subtraction of a Supermartingale Discontinuity 51623. Supermartingale Decompositions and Discontinuities 518

Chapter VLattices and Related Classes of Stochastic Processes 5201. Conventions; The Essential Order 5202. LM x(-) when {*(•), #•(•)} Is a Submartingale 521

Contents XV

3. Uniformly Integrable Positive Submartingales 5234. L" Bounded Stochastic Processes (p > 1) 5245. The Lattices ('S*, £ ) , ( 'S+ , <;), (S±, <0, (S+ , £ ) 5256. The Vector Lattices ('S, <) and (S, <) 5287. The Vector Lattices ('Sm, <) and (Sm, <) 5298. The Vector Lattices ('S,,, <) and (Sp, <) 5309. The Vector Lattices ('S,,,, <) and (S,6, <) 531

10. The Vector Lattices ('Ss, <) and (Ss, <) 53211. The Orthogonal Decompositions 'Sm = 'Smqb + 'Sms and Sm = S^^ + Sms. 53312. Local Martingales and Singular Supermartingale Potentials in (S, <) . . . . 53413. Quasimartingales (Continuous Parameter Context) 535

Chapter VIMarkov Processes 539

1. The Markov Property 5392. Choice of Filtration 5443. Integral Parameter Markov Processes with Stationary Transition Proba-

bilities 5454. Application of Martingale Theory to Discrete Parameter Markov

Processes 5475. Continuous Parameter Markov Processes with Stationary Transition

Probabilities 5506. Specialization to Right Continuous Processes 5527. Continuous Parameter Markov Processes: Lifetimes and Trap Points . . . . 5548. Right Continuity of Markov Process Filtrations; A Zero-One (0-1) Law.. 5569. Strong Markov Property 557

10. Probabilistic Potential Theory; Excessive Functions 56011. Excessive Functions and Supermartingales 56412. Excessive Functions and the Hitting Times of Analytic Sets (Notation and

Hypotheses of Section 11) 56513. Conditioned Markov Processes 56614. Tied Down Markov Processes 56715. Killed Markov Processes 568

Chapter VIIBrownian Motion 570

1. Processes with Independent Increments and State Space UN 5702. Brownian Motion 5723. Continuity of Brownian Paths 5764. Brownian Motion Filtrations 5785. Elementary Properties of the Brownian Transition Density and Brownian

Motion 5816. The Zero-One Law for Brownian Motion 5837. Tied Down Brownian Motion 5868. Andre Reflection Principle 5879. Brownian Motion in an Open Set (N > 1) 589

10. Space-Time Brownian Motion in an Open Set 59211. Brownian Motion in an Interval 594

XVI Contents

12. Probabilistic Evaluation of Parabolic Measure for an Interval 59513. Probabilistic Significance of the Heat Equation and Its Dual • 596

Chapter VIIIThe Ito Integral 599

1. Notation 5992. The Size of Fo 6013. Properties of the Ito Integral 6024. The Stochastic Integral for an Integrand Process in r 0 6055. The Stochastic Integral for an Integrand Process in F 6066. Proofs of the Properties in Section 3 6077. Extension to Vector-Valued and Complex-Valued Integrands 6118. Martingales Relative to Brownian Motion Filtrations 6129. A Change of Variables 615

10. The Role of Brownian Motion Increments 61811. (N = 1) Computation of the Ito Integral by Riemann-Stieltjes Sums 62012. Ito's Lemma 62113. The Composition of the Basic Functions of Potential Theory with Brownian

Motion 62514. The Composition of an Analytic Function with Brownian Motion 626

Chapter IXBrownian Motion and Martingale Theory 627

1. Elementary Martingale Applications 6272. Coparabolic Polynomials and Martingale Theory 6303. Superharmonic and Harmonic Functions on RN and Supermartingales and

Martingales 6324. Hitting of an Fa Set 6355. The Hitting of a Set by Brownian Motion 6366. Superharmonic Functions, Excessive for Brownian Motion 6377. Preliminary Treatment of the Composition of a Superharmonic Function

with Brownian Motion; A Probabilistic Fatou Boundary Limit Theorem . 6418. Excessive and Invariant Functions for Brownian Motion 6459. Application to Hitting Probabilities and to Parabolicity of Transition

Densities 64710. (N = 2). The Hitting of Nonpolar Sets by Brownian Motion 64811. Continuity of the Composition of a Function with Brownian Motion . . . . 64912. Continuity of Superharmonic Functions on Brownian Motion 65013. Preliminary Probabilistic Solution of the Classical Dirichlet Problem . . . . 65114. Probabilistic Evaluation of Reductions 65315. Probabilistic Description of the Fine Topology 65616. a-Excessive Functions for Brownian Motion and Their Composition with

Brownian Motions 65917. Brownian Motion Transition Functions as Green Functions; The Corre-

sponding Backward and Forward Parabolic Equations 66118. Excessive Measures for Brownian Motion 66319. Nearly Borel Sets for Brownian Motion 66620. Brownian Motion into a Set from an Irregular Boundary Point 666

Contents xvn

Chapter XConditional Brownian Motion 668

1. Definition 6682. //-Brownian Motion in Terms of Brownian Motion 6713. Contexts for (2.1) 6764. Asymptotic Character of//-Brownian Paths at Their Lifetimes 6775. //-Brownian Motion from an Infinity of h 6806. Brownian Motion under Time Reversal 6827. Preliminary Probabilistic Solution of the Dirichlet Problem for //-Harmonic

Functions; //-Brownian Motion Hitting Probabilities and theCorresponding Generalized Reductions 684

8. Probabilistic Boundary Limit and Internal Limit Theorems for Ratios ofStrictly Positive Superharmonic Functions 688

9. Conditional Brownian Motion in a Ball 69110. Conditional Brownian Motion Last Hitting Distributions; The Capacitary

Distribution of a Set in Terms of a Last Hitting Distribution 69311. The Tail a Algebra of a Conditional Brownian Motion 69412. Conditional Space-Time Brownian Motion 69913. [Space-Time] Brownian Motion in [RN] UN with Parameter Set R 700

Part 3

Chapter ILattices in Classical Potential Theory and Martingale Theory 7051. Correspondence between Classical Potential Theory and Martingale

Theory 7052. Relations between Decomposition Components of S in Potential Theory

and Martingale Theory 7063. The Classes L" and D 7064. PWB-Related Conditions on //-Harmonic Functions and on Martingales . 7075. Class D Property versus Quasi-Boundedness 7086. A Condition for Quasi-Boundedness 7097. Singularity of an Element of S* 7108. The Singular Component of an Element of S+ 7119. The Class Sp,6 712

10. The Class Sps 71411. Lattice Theoretic Analysis of the Composition of an //-Superharmonic

Function with an //-Brownian Motion 71512. A Decomposition of S*s (Potential Theory Context) 71613. Continuation of Section 11 717

Chapter IIBrownian Motion and the PWB Method 719

1. Context of the Problem 7192. Probabilistic Analysis of the PWB Method 720

XV111 Contents

3. PWB* Examples 7234. Tail a Algebras in the PWB* Context 725

Chapter IIIBrownian Motion on the Mart in Space 727

1. The Structure of Brownian Motion on the Martin Space 7272. Brownian Motions from Martin Boundary Points (Notation of Section 1) 7283. The Zero-One Law at a Minimal Martin Boundary Point and the

Probabilistic Formulation of the Minimal-Fine Topology (Notation ofSection 1) 730

4. The Probabilistic Fatou Theorem on the Martin Space 7325. Probabilistic Approach to Theorem l.XI.4(c) and Its Boundary

Counterparts 7336. Martin Representation of Harmonic Functions in the Parabolic Context . 735

Appendixes

Appendix IAnalytic Sets 741

1. Pavings and Algebras of Sets 7412. Suslin Schemes 7413. Sets Analytic over a Product Paving 7424. Analytic Extensions versus a Algebra Extensions of Pavings 7435. Projection Characterization s?(<&) 7436. The Operation s4{s4) 7447. Projections of Sets in Product Pavings 7448. Extension of a Measurability Concept to the Analytic Operation Context. 7459. The Gs Sets of a Complete Metric Space 745

10. Polish Spaces 74611. The Baire Null Space 74612. Analytic Sets 74713. Analytic Subsets of Polish Spaces 748

Appendix IICapacity Theory 750

1. Choquet Capacities 7502. Sierpinski Lemma 7503. Choquet Capacity Theorem 7514. Lusin's Theorem 7515. A Fundamental Example of a Choquet Capacity 7526. Strongly Subadditive Set Functions 7527. Generation of a Choquet Capacity by a Positive Strongly Subadditive Set

Function 7538. Topological Precapacities 7559. Universally Measurable Sets 756

Contents xix

Appendix IIILattice Theory 758

1. Introduction 7582. Lattice Definitions 7583. Cones 7584. The Specific Order Generated by a Cone 7595. Vector Lattices 7606. Decomposition Property of a Vector Lattice 7627. Orthogonality in a Vector Lattice 7628. Bands in a Vector Lattice 7629. Projections on Bands 763

10. The Orthogonal Complement of a Set 76411. The Band Generated by a Single Element 76412. Order Convergence 76513. Order Convergence on a Linearly Ordered Set 766

Appendix IVLattice Theoretic Concepts in Measure Theory 767

1. Lattices of Set Algebras 7672. Measurable Spaces and Measurable Functions 7673. Composition of Functions 7684. The Measure Lattice of a Measurable Space 7695. The a Finite Measure Lattice of a Measurable Space (Notation of Section 4) 7716. The Hahn and Jordan Decompositions 7727. The Vector Lattice Ma 7728. Absolute Continuity and Singularity 7739. Lattices of Measurable Functions on a Measure Space 774

10. Order Convergence of Families of Measurable Functions 77511. Measures on Polish Spaces 77712. Derivates of Measures 778

Appendix VUniform Integrability 779

Appendix VIKernels and Transition Functions 781

1. Kernels 7812. Universally Measurable Extension of a Kernel 7823. Transition Functions 782

Appendix VIIIntegral Limit Theorems 785

1. An Elementary Limit Theorem 7852. Ratio Integral Limit Theorems 7863. A One-Dimensional Ratio Integral Limit Theorem 7864. A Ratio Integral Limit Theorem Involving Convex Variational Derivates. 788

XX Contents

Appendix VIIILower Semicontinuous Functions 791

1. The Lower Semicontinuous Smoothing of a Function 7912. Suprema of Families of Lower Semicontinuous Functions 7913. Choquet Topological Lemma 792

Historical Notes 793Part 1 793Part 2 806Part 3 815Appendixes 816

Bibliography 819Notation Index 827Index 829