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Part 1 Classical and Parabolic Potential Theory 3

Chapter Ⅰ Introduction to the Mathematical Background of Classical Potential Theory 3

1．The Context of Green's Identity 3

2．Function Averages 4

3．Harmonic Functions 4

4．Maximum-Minimum Theorem for Harmonic Functions 5

5．The Fundamental Kernel for RN and Its Potentials 6

6．Gauss Integral Theorem 7

7．The Smoothness of Potentials;The Poisson Equation 8

8．Harmonic Measure and the Riesz Decomposition 11

Chapter Ⅱ Basic Properties of Harmonic,Subharmonic,and Superharmonic Functions 14

1．The Green Function of a Ball;The Poisson Integral 14

2．Harnack's Inequality 16

3．Convergence of Directed Sets of Harmonic Functions 17

4．Harmonic,Subharmonic,and Superharmonic Functions 18

5．Minimum Theorem for Superharmonic Functions 20

6．Application of the Operation τB 20

7．Characterization of Superharmonic Functions in Terms of Harmonic Functions 22

8．Differentiable Superharmonic Functions 23

9．Application of Jensen's Inequality 23

10．Superharmonic Functions on an Annulus 24

11．Examples 25

12．The Kelvin Transformation(N≥2) 26

13．Greenian Sets 27

14 The L1(μB-)and D(μB-)Classes of Harmonic Functions on a Ball B;The Riesz-Herglotz Theorem 27

15．The Fatou Boundary Limit Theorem 31

16．Minimal Harmonic Functions 33

Chapter Ⅲ Infima of Families of Superharmonic Functions 35

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

2．Generalization of Theorem 1 36

3．Fundamental Convergence Theorem(Preliminary Version) 37

4．The Reduction Operation 38

5．Reduction Properties 41

6．A Smallness Property of Reductions on Compact Sets 42

7．The Natural(Pointwise)Order Decomposition for Positive Superharmonic Functions 43

Chapter Ⅳ Potentials on Special Open Sets 45

1．Special Open Sets,and Potentials on Them 45

2．Examples 47

3．A Fundamental Smallness Property of Potentials 48

4．Increasing Sequences of Potentials 49

5．Smoothing of a Potential 49

6．Uniqueness of the Measure Determining a Potential 50

7．Riesz Measure Associated with a Superharmonic Function 51

8．Riesz Decomposition Theorem 52

9．Counterpart for Superharmonic Functions on R2 of the Riesz Decomposition 53

10 An Approximation Theorem 55

Chapter Ⅴ Polar Sets and Their Applications 57

1．Definition 57

2．Superharmonic Functions Associated with a Polar Set 58

3．Countable Unions of Polar Sets 59

4．Properties of Polar Sets 59

5．Extension of a Superharmonic Function 60

6．Greenian Sets in R2 as the Complements of Nonpolar Sets 63

7．Superharmonic Function Minimum Theorem(Extension of Theorem II.5) 63

8．Evans-Vasilesco Theorem 64

9．Approximation of a Potential by Continuous Potentials 66

10．The Domination Principle 67

11．The Infinity Set of a Potential and the Riesz Measure 68

Chapter Ⅵ The Fundamental Convergence Theorem and the Reduction Operation 70

1．The Fundamental Convergence Theorem 70

2．Inner Polar versus Polar Sets 71

3．Properties of the Reduction Operation 74

4．Proofs of the Reduction Properties 77

5．Reductions and Capacities 84

Chapter Ⅶ Green Functions 85

1．Definition of the Green Function GD 85

2．Extremal Property of GD 87

3．Boundedness Properties of GD 88

4．Further Properties of GD 90

5．The Potential GDμ of a Measure μ 92

6．Increasing Sequences of Open Sets and the Corresponding Green Function Sequences 94

7．The Existence of GD versus the Greenian Character of D 94

8．From Special to Greenian Sets 95

9．Approximation Lemma 95

10．The Function GD(·,ζ)|D-{ζ} as a Minimal Harmonic Function 96

Chapter Ⅷ The Dirichlet Problem for Relative Harmonic Functions 98

1．Relative Harmonic,Superharmonic,and Subharmonic Functions 98

2．The PWB Method 99

3．Examples 104

4．Continuous Boundary Functions on the Euclidean Boundary(h≡1) 106

5．h-Harmonic Measure Null Sets 108

6．Properties of PWBh Solutions 110

7．Proofs for Section 6 111

8．h-Harmonic Measure 114

9．h-Resolutive Boundaries 118

10．Relations between Reductions and Dirichlet Solutions 122

11．Generalization of the Operator τh B and Application to GMh 123

12．Barriers 124

13．h-Barriers and Boundary Point h-Regularity 126

14．Barriers and Euclidean Boundary Point Regularity 127

15．The Geometrical Significance of Regularity(Euclidean Boundary,h≡1) 128

16．Continuation of Section 13 130

17．h-Harmonic Measure μh D as a Function ofD 131

18．The Extension G＝ D of GD and the Harmonic Average μD(ξ,G＝ B(η,·))When D ？ B 132

19．Modification of Section 18 for D=R2 136

20．Interpretation of φD as a Green Function with Pole ∞(N=2) 139

21．Variant of the Operator τB 140

Chapter Ⅸ Lattices and Related Classes of Functions 141

1．Introduction 141

2．LMh D u for an h-Subharmonic Function u 141

3．The Class D(μh D-) 142

4．The Class Lp(μh D-)(p≥1) 144

5．The Lattices(S±,≤)and(S+,≤) 145

6．The Vector Lattice (S,≤) 146

7．The Vector Lattice Sm 148

8．The Vector Lattice Sp 149

9．The Vector Lattice Sqb 150

10．The Vector Lattice Ss 151

11．A Refinement ofthe Riesz Decomposition 152

12．Lattices of h-Harmonic Functions on a Ball 152

Chapter Ⅹ The Sweeping Operation 155

1．Sweeping Context and Terminology 155

2． Relation between Harmonic Measure and the Sweeping Kernel 157

3．Sweeping Symmetry Theorem 158

4．Kernel Property of δA D 158

5．Swept Measures and Functions 160

6．Some Properties of δA D 161

7．Poles of a Positive Harmonic Function 163

8．Relative Harmonic Measure on a Polar Set 164

Chapter Ⅺ The Fine Topology 166

1．Definitions and Basic Properties 166

2．A Thinness Criterion 168

3．Conditions That ξ∈A∫ 169

4．An Internal Limit Theorem 171

5．Extension of the Fine Topology to RN∪{∞} 175

6．The Fine Topology Derived Set of a Subset of RN 177

7．Application to the Fundamental Convergence Theorem and to Reductions 177

8．Fine Topology Limits and Euclidean Topology Limits 178

9．Fine Topology Limits and Euclidean Topology Limits(Continued) 179

10．Identification of A∫ in Terms ofa Special Functionu 180

11．Quasi-Lindel？f Property 180

12．Regularity in Terms of the Fine Topology 181

13．The Euclidean Boundary Set of Thinness of a Greenian Set 182

14．The Support of a Swept Measure 183

15．Characterization of ‖μ‖A 183

16．A Special Reduction 184

17．The Fine Interior of a Set of Constancy of a Superharmonic Function 184

18．The Support of a Swept Measure (Continuation of Section 14) 185

19．Superharmonic Functions on Fine-Open Sets 187

20．A Generalized Reduction 187

21．Limits of Superharmonic Functions at Irregular Boundary Points of Their Domains 190

22．The Limit Harmonic Measure ∫μD 191

23．Extension of the Domination Principle 194

Chapter Ⅻ The Martin Boundary 195

1．Motivation 195

2．The Martin Functions 196

3．The Martin Space 197

4．Preliminary Representations of Positive Harmonic Functions and Their Reductions 199

5．Minimal Harmonic Functions and Their Poles 200

6．Extension of Lemma 4 201

7．The Set of Nonminimal Martin Boundary Points 202

8．Reductions on the Set of Minimal Martin Boundary Points 203

9．The Martin Representation 204

10．Resolutivity of the Martin Boundary 207

11．Minimal Thinness at a Martin Boundary Point 208

12．The Minimal-Fine Topology 210

13．First Martin Boundary Counterpart of Theorem XI.4(c)and(d) 213

14．Second Martin Boundary Counterpart of Theorem XI.4(c) 213

15．Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point 215

16．Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point(Continued) 216

17．Minimal-Fine Martin Boundary Limit Functions 216

18．The Fine Boundary Function of a Potential 218

19．The Fatou Boundary Limit Theorem for the Martin Space 219

20．Classical versus Minimal-Fine Topology Boundary Limit Theorems for Relative Superharmonic Functions on a Ball in RN 221

21．Nontangential and Minimal-Fine Limits at a Half-space Boundary 222

22．Normal Boundary Limits for a Half-space 223

23．Boundary Limit Function (Minimal-Fine and Normal)of a Potential on a Half-space 225

Chapter ⅩⅢ Classical Energy and Capacity 226

1．Physical Context 226

2．Measures and Their Energies 227

3．Charges and Their Energies 228

4．Inequalities between Potentials,and the Corresponding Energy Inequalities 229

5．The Function D？GDμ 230

6．Classical Evaluation of Energy;Hilbert Space Methods 231

7．The Energy Functional(Relative to an Arbitrary Greenian Subset D of RN) 233

8．Alternative Proofs ofTheorem 7(b+) 235

9．Sharpening of Lemma 4 237

10．The Classical Capacity Function 237

11．Inner and Outer Capacities(Notation of Section 10) 240

12．Extremal Property Characterizations of Equilibrium Potentials(Notation of Section 10) 241

13．Expressions for C(A) 243

14．The Gauss Minimum Problems and Their Relation to Reductions 244

15．Dependence of C on D 247

16．Energy Relative to R2 248

17．The Wiener Thinness Criterion 249

18．The Robin Constant and Equilibrium Measures Relative to R2(N=2) 251

Chapter ⅩⅣ One-Dimensional Potential Theory 256

1．Introduction 256

2．Harmonic,Superharmonic,and Subharmonic Functions 256

3．Convergence Theorems 256

4．Smoothness Properties of Superharmonic and Subharmonic Functions 257

5．The Dirichlet Problem(Euclidean Boundary) 257

6．Green Functions 258

7．Potentials of Measures 259

8．Identification of the Measure Defining a Potential 259

9．Riesz Decomposition 260

10．The Martin Boundary 261

Chapter ⅩⅤ Parabolic Potential Theory:Basic Facts 262

1．Conventions 262

2．The Parabolic and Coparabolic Operators 263

3．Coparabolic Polynomials 264

4．The Parabolic Green Function of RN 266

5．Maximum-Minimum Parabolic Function Theorem 267

6．Application of Green's Theorem 269

7．The Parabolic Green Function of a Smooth Domain;The Riesz Decomposition and Parabolic Measure(Formal Treatment) 270

8．The Green Function of an Interval 272

9．Parabolic Measure for an Interval 273

10．Parabolic Averages 275

11．Harnack's Theorems in the Parabolic Context 276

12．Superparabolic Functions 277

13．Superparabolic Function Minimum Theorem 279

14．The Operation ？ and the Defining Average Properties of Superparabolic Functions 280

15．Superparabolic and Parabolic Functions on a Cylinder 281

16．The Appell Transformation 282

17．Extensions of a Parabolic Function Defined on a Cylinder 283

Chapter ⅩⅥ Subparabolic,Superparabolic,and Parabolic Functions on a Slab 285

1．The Parabolic Poisson Integral fora Slab 285

2．A Generalized Superparabolic Function Inequality 287

3．A Crrterion of a Subparabolic Function Supremum 288

4．A Boundary Limit Criterion for the Identically Vanishing of a Positive Parabolic Function 288

5．A Condition that a Positive Parabolic Function Be Representable by a Poisson Integral 290

6．The L1(？-)and D(？-)Classes of Parabolic Functions on a Slab 290

7．The Parabolic Boundary Limit Theorem 292

8．Minimal Parabolic Functions on a Slab 293

Chapter ⅩⅦ Parabolic Potential Theory (Continued) 295

1．Greatest Minorants and Least Majorants 295

2．The Parabolic Fundamental Convergence Theorem(Preliminary Version)and the Reduction Operation 295

3．The Parabolic Context Reduction Operations 296

4．The Parabolic Green Function 298

5．Potentials 300

6．The Smoothness of Potentials 303

7．Riesz Decomposition Theorem 305

8．Parabolic-Polar Sets 305

9．The Parabolic-Fine Topology 308

10．Semipolar Sets 309

11．Preliminary List of Reduction Properties 310

12．A Criterion of Parabolic Thinness 313

13．The Parabolic Fundamental Convergence Theorem 314

14．Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions 316

15．Applications of the Fundamental Convergence Theorem to the Parabolic-Fine Topology 317

16．Parabolic-Reduction Properties 317

17．Proofs of the Reduction Properties in Section 16 320

18．The Classical Context Green Function in Terms of the Parabolic Context Green Function(N≥1) 326

19．The Quasi-Lindel？f Property 328

Chapter ⅩⅧ The Parabolic Dirichlet Problem,Sweeping,and Exceptional Sets 329

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

2．h-Parabolic Measure 332

3．Parabolic Barriers 333

4．Relations between the Classical Dirichlet Problem and the Parabolic Context Diriehlet Problem 334

5．Classical Reductions in the Parabolic Context 335

6．Parabolic Regularity of Boundary Points 337

7．Parabolic Regularity in Terms of the Fine Topology 341

8．Sweeping in the Parabolic Context 341

9．The Extension ？of ？ and the Parabolic Average ？(？,？(·,？)when ？ 343

10．Conditions that ξ∈？ps 345

11．Parabolic-and Coparabolic-Polar Sets 347

12．Parabolic-and Coparabolic-Semipolar Sets 348

13．The Support of a Swept Measure 350

14．An Internal Limit Theorem;The Coparabolic-Fine Topology Smoothness of Superparabolic Functions 351

15．Application to a Version of the Parabolic Context Fatou Boundary Limit Theorem on a Slab 357

16．The Parabolic Context Domination Principle 358

17．Limits of Superparabolic Functions at Parabolic-Irregular Boundary Points of Their Domains 358

18．Martin Flat Point Set Pairs 361

19．Lattices and Related Classes of Functions in the Parabolic Context 361

Chapter ⅩⅨ The Martin Boundary in the Parabolic Context 363

1．Introduction 363

2．The Martin Functions of Martin Point Set and Measure Set Pairs 364

3．The Martin Space ？M 366

4．Preparatory Material for the Parabolic Context Martin Representation Theorem 367

5．Minimal Parabolic Functions and Their Poles 369

6．The Set of Nonminimal Martin Boundary Points 370

7．The Martin Representation in the Parabolic Context 371

8 Martin Boundary of a Slab ？=RN×]0,δ[with 0＜δ≤+∞ 371

9．Martin Boundaries for the Lower Half-space of？Nand for ？N 374

10．The Martin Boundary of？=]0,+∞[×]-∞,δ[ 375

11．？WB？Solutions on ？M 377

12．The Minimal-Fine Topology in the Parabolic Context 377

13．Boundary Counterpart ofTheorem ⅩⅧ.14(f) 379

14．The Vanishing ofPotentials on ？M？ 381

15．The Parabolic Context Fatou Boundary Limit Theorem on Martin Spaces 381

Part 2 Probabilistic Counterpart of Part1 387

Chapter Ⅰ Fundamental Concepts of Probability 387

1．Adapted Families of Functions on Measurable Spaces 387

2．Progressive Measurability 388

3．Random Variables 390

4．Conditional Expectations 391

5．Conditional Expectation Continuity Theorem 393

6．Fatou's Lemma for Conditional Expectations 396

7．Dominated Convergence Theorem for Conditional Expectations 397

8．Stochastic Processes,“Evanescent,""Indistinguishable,""Standard Modification,""Nearly" 398

9．The Hitting of Sets and Progressive Measurability 401

10．Canonical Processes and Finite-Dimensional Distributions 402

11．Choice of the Basic Probability Space 404

12．The Hitting of Sets by a Right Continuous Process 405

13．Measurability versus Progressive Measurability of Stochastic Processes 407

14．Predictable Families of Functions 410

Chapter Ⅱ Optional Times and Associated Concepts 413

1．The Context of Optional Times 413

2．Optional Time Properties(Continuous Parameter Context) 415

3．Process Functions at Optional Times 417

4．Hitting and Entry Times 419

5．Application to Continuity Properties of Sample Functions 421

6．Continuation of Section 5 423

7．Predictable Optional Times 423

8．Section Theorems 425

9．The Graph of a Predictable Time and the Entry Time of a Predictable Set 426

10．Semipolar Subsets of R+×Ω 427

11．The Classes D and Lp of Stochastic Processes 428

12．Decomposition of Optional Times;Accessible and Totally Inaccessible Optional Times 429

Chapter Ⅲ Elements of Martingale Theory 432

1．Definitions 432

2．Examples 433

3．Elementary Properties(Arbitrary Simply Ordered Parameter Set) 435

4．The Parameter Set in Martingale Theory 437

5．Convergence of Supermartingale Families 437

6．Optional Sampling Theorem(Bounded Optional Times) 438

7．Optional Sampling Theorem for Right Closed Processes 440

8．Optional Stopping 442

9．Maximal Inequalities 442

10．Conditional Maximal Inequalities 444

11．An Lp Inequality for Submartingale Suprema 444

12．Crossings 445

13．Forward Convergence in the L1 Bounded Case 450

14．Convergence ofa Uniformly Integrable Martingale 451

15．Forward Convergence of a Right Closable Supermartingale 453

16．Backward Convergence of a Martingale 454

17．Backward Convergence of a Supermartingale 455

18．The τ Operator 455

19．The Natural Order Decomposition Theorem for Supermartingales 457

20．The Operators LM and GM 458

21．Supermartingale Potentials and the Riesz Decomposition 459

22．Potential Theory Reductions in a Discrete Parameter Probability Context 459

23．Application to the Crossing Inequalities 461

Chapter Ⅳ Basic Properties of Continuous Parameter Supermartingales 463

1．Continuity Properties 463

2．Optional Sampling of Uniformly Integrable Continuous Parameter Martingales 468

3．Optional Sampling and Convergence of Continuous Parameter Supermartingales 470

4．Increasing Sequences of Supermartingales 473

5．Probability Version of the Fundamental Convergence Theorem of Potential Theory 476

6．Quasi-Bounded Positive Supermartingales;Generation of Supermartingale Potentials by Increasing Processes 480

7．Natural versus Predictable Increasing Processes(I=Z+ or R+) 483

8．Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case 488

9．An Inequality for Predictable Increasing Processes 489

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

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

12．Meyer Decomposition of a Submartingale 495

13．Role of the Measure Associated with a Supermartingale;The Supermartingale Domination Principle 496

14．The Operators τ,LM,and GM in the Continuous Parameter Context 500

15．Potential Theory on R+×Ω 501

16．The FineTopology of R+× Ω 502

17．Potential Theory Reductions in a Continuous Parameter Probability Context 504

18．Reduction Properties 505

19．Proofs ofthe Reduction Properties in Section 18 509

20．Evaluation of Reductions 513

21．The Energy of a Supermartingale Potential 515

22．The Subtraction of a Supermartingale Discontinuity 516

23．Supermartingale Decompositions and Discontinuities 518

Chapter Ⅴ Lattices and Related Classes of Stochastic Processes 520

1．Conventions;The Essential Order 520

2．LMx(·)when{x(·),F(·)}Is a Submartingale 521

3．Uniformly Integrable Positive Submartingales 523

4．LpBounded Stochastic Processes(p≥1) 524

5．The Lattices('S±,≤),('S+,≤),(S±,≤),(S+,≤) 525

6．The Vector Lattices('S,≤)and(S,≤) 528

7．The Vector Lattices('Sm,≤)and(Sm,≤) 529

8．The Vector Lattices('Sp,≤)and(Sp,≤) 530

9．The Vector Lattices('Sqb,≤)and(Sqb,≤) 531

10．The Vector Lattices('Ss,≤)and(Ss,≤) 532

11．The Orthogonal Decompositions'Sm='Smqb+'Sms and Sm=Smqb+Sms 533

12．Local Martingales and Singular Supermartingale Potentials in(S,≤) 534

13．Quasimartingales(Continuous Parameter Context) 535

Chapter Ⅵ Markov Processes 539

1．The Markov Property 539

2．Choice of Filtration 544

3．Integral Parameter Markov Processes with Stationary Transition Probabilities 545

4．Application of Martingale Theory to Discrete Parameter Markov Processes 547

5．Continuous Parameter Markov Processes with Stationary Transition Probabilities 550

6．Specialization to Right Continuous Processcs 552

7．Continuous Parameter Markov Processes:Lifctimes and Trap Points 554

8．Right Continuity of Markov Process Filtrations;A Zero-One(0-1)Law 556

9．Strong Markov Property 557

10．Probabilistic Potential Theory;Excessive Functions 560

11．Excessive Functions and Supermartingales 564

12．Excessive Functions and the Hitting Times of Analytic Sets(Notation and Hypotheses of Section 11) 565

13．Conditioned Markov Processes 566

14．Tied Down Markov Processes 567

15．Killed Markov Processes 568

Chapter Ⅵ Brownian Motion 570

1．Processes with Independent Increments and State Space RN 570

2．Brownian Motion 572

3．Continuity of Brownian Paths 576

4．Brownian Motion Filtrations 578

5．Elementary Properties of the Brownian Transition Density and Brownian Motion 581

6．The Zero-One Law for Brownian Motion 583

7．Tied Down Brownian Motion 586

8．André Reflection Principle 587

9．Brownian Motion in an Open Set(N≥1) 589

10．Space-Time Brownian Motion in an Open Set 592

11．Brownian Motion in an Intervai 594

12．Probabilistic Evaluation of Parabolic Measure for an Interval 595

13．Probabilistic Significance of the Heat Equation and Its Dual 596

Chapter Ⅷ 599

The It？ Integral 599

1．Notation 599

2．The Size of Г0 601

3．Properties ofthe It？ Integral 602

4．The Stochastic Integral for an Integrand Process in Г0 605

5．The Stochastic Integral for an Integrand Process in Г 606

6．Proofs of the Properties in Section 3 607

7．Extension to Vector-Valued and Complex-Valued Integrands 611

8．Martingales Relative to Brownian Motion Filtrations 612

9．A Change of Variables 615

10．The Role of Brownian Motion Increments 618

11．(N=1)Computation of the It？ Integral by Riemann-Stieltjes Sums 620

12．It？'s Lemma 621

13．The Composition of the Basic Functions of Potential Theory with Brownian Motion 625

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

Chapter Ⅸ Brownian Motion and Martingale Theory 627

1．Elementary Martingale Applications 627

2．Coparabolic Polynomials and Martingale Theory 630

3．Superharmonic and Harmonic Functions on RN and Supermartingales and Martingales 632

4．Hitting of an Fσ Set 635

5．The Hitting of a Set by Brownian Motion 636

6．Superharmonic Functions,Excessive for Brownian Motion 637

7．Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion;A Probabilistic Fatou Boundary Limit Theorem 641

8．Excessive and Invariant Functions for Brownian Motion 645

9．Application to Hitting Probabilities and to Parabolicity of Transition Densities 647

10．(N=2).The Hitting of Nonpolar Sets by Brownian Motion 648

11．Continuity of the Composition of a Function with Brownian Motion 649

12．Continuity of Superharmonic Functions on Brownian Motion 650

13．Preliminary Probabilistic Solution of the Classical Dirichlet Problem 651

14．Probabilistic Evaluation of Reductions 653

15．Probabilistic Description of the Fine Topology 656

16．α-Excessive Functions for Brownian Motion and Their Composition with Brownian Motions 659

17．Brownian Motion Transition Functions as Green Functions;The Corresponding Backward and Forward Parabolic Equations 661

18．Excessive Measures for Brownian Motion 663

19．Nearly Borel Sets for Brownian Motion 666

20．Brownian Motion into a Set from an Irregular Boundary Point 666

Chapter Ⅹ Conditional Brownian Motion 668

1．Definition 668

2．h-Brownian Motion in Terms of Brownian Motion 671

3．Contexts for(2.1) 676

4．Asymptotic Character of h-Brownian Paths at Their Lifetimes 677

5．h-Brownian Motion from an Infinity of h 680

6．Brownian Motion under Time Reversal 682

7．Preliminary Probabilistic Solution of the Dirichlet Problem for h-Harmonic Functions;h-Brownian Motion Hitting Probabilities and the Corresponding Generalized Reductions 684

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

9．Conditional Brownian Motion in a Ball 691

10．Conditional Brownian Motion Last Hitting Distributions;The Capacitary Distribution of a Set in Terms of a Last Hitting Distribution 693

11．The TailσAlgebra ofa Conditional Brownian Motion 694

12．Conditional Space-Time Brownian Motion 699

13．[Space-Time]Brownian Motion in[？N]RN with Parameter Set R 700

Part3 705

Chapter Ⅰ Lattices in Classical Potential Theory and Martingale Theory 705

1．Correspondence between Classical Potential Theory and Martingale Theory 705

2．Relations between Decomposition Components of S in Potential Theory and Martingale Theory 706

3．The Classes Lp and D 706

4．PWB-Related Conditions on h-Harmonic Functions and on Martingales 707

5．Class D Property versus Quasi-Boundedness 708

6．A Condition for Quasi-Boundedness 709

7．Singularity of an Element of S+ m 710

8．The Singular Component of an Element of S+ 711

9．The Class Spqb 712

10．The Class Sps 714

11．Lattice Theoretic Analysis of the Composition of an h-Superharmonic Function with an h-Brownian Motion 715

12．A Decomposition of S+ ms(Potential Theory Context) 716

13．Continuation of Section 11 717

Chapter Ⅱ Brownian Motion and the PWB Method 719

1．Context of the Problem 719

2．Probabilistic Analysis of the PWB Method 720

3．PWBh Examples 723

4．Tail σ Algebras in the PWBh Context 725

Chapter Ⅲ Brownian Motion on the Martin Space 727

1．The Structure of Brownian Motion on the Martin Space 727

2．Brownian Motions from Martin Boundary Points(Notation of Section 1) 728

3．The Zero-One Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the Minimal-Fine Topology(Notation of Section 1) 730

4．The Probabilistic Fatou Theorem on the Martin Space 732

5．Probabilistic Approach to Theorem 1.XI.4(c)and Its Boundary Counterparts 733

6．Martin Representation of Harmonic Functions in the Parabolic Context 735

Appendixes 741

Appendix Ⅰ Analytic Sets 741

1．Pavings and Algebras of Sets 741

2．Suslin Schemes 741

3．Sets Analytic over a Product Paving 742

4．Analytic Extensions versus σ Algebra Extensions of Pavings 743

5．Projection Characterization A(y) 743

6．The Operation A(A) 744

7．Projections of Sets in Product Pavings 744

8．Extension of a Measurability Concept to the Analytic Operation Context 745

9．The Gδ Sets of a Complete Metric Space 745

10．Polish Spaces 746

11．The Baire Null Space 746

12．Analytic Sets 747

13．Analytic Subsets of Polish Spaces 748

Appendix Ⅱ Capacity Theory 750

1．Choquet Capacities 750

2．Sierpinski Lemma 750

3．Choquet Capacity Theorem 751

4．Lusin's Theorem 751

5．A Fundamental Example of a Choquet Capacity 752

6．Strongly Subadditive Set Functions 752

7．Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function 753

8．Topological Precapacities 755

9．Universally Measurable Sets 756

Appendix Ⅲ Lattice Theory 758

1．Introduction 758

2．Lattice Definitions 758

3．Cones 758

4．The Specific Order Generated by a Cone 759

5．Vector Lattices 760

6．Decomposition Property of a Vector Lattice 762

7．Orthogonality in a Vector Lattice 762

8．Bands in a Vector Lattice 762

9．Projections on Bands 763

10．The Orthogonal Complement of a Set 764

11．The Band Generated by a Single Element 764

12．Order Convergence 765

13．Order Convergence on a Linearly Ordered Set 766

Appendix Ⅳ Lattice Theoretic Concepts in Measure Theory 767

1．Lattices of Set Algebras 767

2．Measurable Spaces and Measurable Functions 767

3．Composition of Functions 768

4．The Measure Lattice of a Measurable Space 769

5．The σ Finite Measure Lattice of a Measurable Space(Notation of Section 4) 771

6．The Hahn and Jordan Decompositions 772

7．The Vector Lattice Mσ 772

8．Absolute Continuity and Singularity 773

9．Lattices of Measurable Functions on a Measure Space 774

10．Order Convergence of Families of Measurable Functions 775

11．Measures on Polish Spaces 777

12．Derivates of Measures 778

Appendix Ⅴ Uniform Integrability 779

Appendix Ⅵ Kernels and Transition Functions 781

1．Kernels 781

2．Universally Measurable Extension of a Kernel 782

3．Transition Functions 782

Appendix Ⅶ Integral Limit Theorerns 785

1．An Elementary Limit Theorem 785

2．Ratio Integral Limit Theorems 786

3．A One-Dimensional Ratio Integral Limit Theorem 786

4．A Ratio Integral Limit Theorem Involving Convex Variational Derivates 788

Appendix Ⅷ Lower Semicontinuous Functions 791

1．The Lower Sernicontinuous Smoothing of a Function 791

2．Suprema of Families of Lower Semicontinuous Functions 791

3．Choquet Topological Lemma 792

Historical Notes 793

Part 1 793

Part 2 806

Part 3 815

Appendixes 816

Bibliography 819

Notation Index 827

Index 829