Classical potential theory and its probabilistic counterpart

Contents......Page 8
Introduction......Page 24
Notation and Conventions......Page 28
Part 1 Classical and Parabolic Potential Theory ......Page 30
1. The Context of Green's Identity ......Page 32
3. Harmonic Functions ......Page 33
4. Maximum-Minimum Theorem for Harmonic Functions ......Page 34
5. The Fundamental Kernel for R" and Its Potentials ......Page 35
6. Gauss Integral Theorem ......Page 36
7. The Smoothness of Potentials; The Poisson Equation ......Page 37
8. Harmonic Measure and the Riesz Decomposition ......Page 40
1. The Green Function of a Ball; The Poisson Integral ......Page 43
2. Harnack's Inequality ......Page 45
3. Convergence of Directed Sets of Harmonic Functions ......Page 46
4. Harmonic, Subharmonic, and Superharmonic Functions ......Page 47
6. Application of the Operation tao_B ......Page 49
7. Characterization of Superharmonic Functions in Terms of Harmonic Functions ......Page 51
9. Application of Jensen's Inequality ......Page 52
10. Superharmonic Functions on an Annulus ......Page 53
11. Examples ......Page 54
12. The Kelvin Transformation (N>2) ......Page 55
14. The L' (? ) and D(? _) Classes of Harmonic Functions on a Ball B: The Riesz Herglotz Theorem ......Page 56
15 The Fatou Boundary Limit Theorem ......Page 60
16. Minimal Harmonic Functions ......Page 62
1. Least Superharmonic Majorant (LM) and Greatest Subharmonic Minorant (GM) ......Page 64
2. Generalization of Theorem I ......Page 65
3 Fundamental Convergence Theorem (Preliminary Version) ......Page 66
4. The Reduction Operation ......Page 67
5. Reduction Properties ......Page 70
6. A Smallness Property of Reductions on Compact Sets ......Page 71
7 The Natural (Pointwise) Order Decomposition for Positive Superharmonic Functions ......Page 72
1. Special Open Sets, and Potentials on Them ......Page 74
2. Examples ......Page 76
3. A Fundamental Smallness Property of Potentials ......Page 77
5. Smoothing of a Potential ......Page 78
6. Uniqueness of the Measure Determining a Potential ......Page 79
7. Riesz Measure Associated with a Superharmonic Function ......Page 80
8. Riesz Decomposition Theorem ......Page 81
9. Counterpart for Superharmonic Functions on R' of the Riesz Decomposition ......Page 82
10. An Approximation Theorem ......Page 84
1. Definition ......Page 86
2. Superharmonic Functions Associated with a Polar Set ......Page 87
4. Properties of Polar Sets ......Page 88
5. Extension of a Superharmonic Function ......Page 89
7. Superharmonic Function Minimum Theorem (Extension of Theorem 11.5) ......Page 92
8. Evans- Vasilesco Theorem ......Page 93
9. Approximation of a Potential by Continuous Potentials ......Page 95
10 The Domination Principle ......Page 96
11 The Infinity Set of a Potential and the Riesz Measure ......Page 97
1. The Fundamental Convergence Theorem ......Page 99
2. Inner Polar versus Polar Sets ......Page 100
3. Properties of the Reduction Operation ......Page 103
4. Proofs of the Reduction Properties ......Page 106
5. Reductions and Capacities ......Page 113
1. Definition of the Green Function G_D ......Page 114
2. Extremal Property of G_D ......Page 116
3. Boundedness Properties of G_D ......Page 117
4. Further Properties of G_D ......Page 119
5. The Potential G_Dnu of a Measure nu ......Page 121
7. The Existence of G_D versus the Greenian Character of D ......Page 123
9. Approximation Lemma ......Page 124
10. The Function G_D(.,xi)_{|D-xi|}, as a Minimal Harmonic Function ......Page 125
1. Relative Harmonic, Superharmonic, and Subharmonic Functions ......Page 127
2. The PWB Method ......Page 128
3. Examples ......Page 133
4. Continuous Boundary Functions on the Euclidean Boundary (h =1) ......Page 135
5. h-Harmonic Measure Null Sets ......Page 137
6. Properties of PWB^h Solutions ......Page 139
7. Proofs for Section 6 ......Page 140
8. h-Harmonic Measure ......Page 143
9. h-Resolutive Boundaries ......Page 147
10. Relations between Reductions and Dirichlet Solutions ......Page 151
11. Generalization of the Operator rB and Application to GM" ......Page 152
12. Barriers ......Page 153
13. h-Barriers and Boundary Point h-Regularity ......Page 155
14. Barriers and Euclidean Boundary Point Regularity ......Page 156
15. The Geometrical Significance of Regularity (Euclidean Boundary, h = 1) ......Page 157
16. Continuation of Section 13 ......Page 159
17. h-Harmonic Measure nu_D^h as a Function of D ......Page 160
18. The Extension G_D^= of G_D and the Harmonic Average G8 011 9) When D c B ......Page 161
19. Modification of Section 18 for D = 682 ......Page 165
20. Interpretation of 0D as a Green Function with Pole oo (N = 2) ......Page 168
21. Variant of the OperatortB ......Page 169
2. LMD u for an h-Subharmonic Function a ......Page 170
3. The Class D(Akl)-) ......Page 171
4. The Class L"(?-)(P ? I) ......Page 173
5. The Lattices (Si, 5) and (S`, E) ......Page 174
6. The Vector Lattice (S. <) ......Page 175
7. The Vector Lattice S ......Page 177
8. The Vector Lattice SD ......Page 178
9. The Vector Lattice Sqb ......Page 179
10. The Vector Lattice S. ......Page 180
12. Lattices of h-Harmonic Functions on a Ball ......Page 181
1. Sweeping Context and Terminology ......Page 184
2. Relation between Harmonic Measure and the Sweeping Kernel ......Page 186
4. Kernel Property of Sr; ......Page 187
5. Swept Measures and Functions ......Page 189
6. Some Properties of Si; ......Page 190
7. Poles of a Positive Harmonic Function ......Page 192
8. Relative Harmonic Measure on a Polar Set ......Page 193
1. Definitions and Basic Properties ......Page 195
2. A Thinness Criterion ......Page 197
3. Conditions That I,' e Af ......Page 198
4. An Internal Limit Theorem ......Page 200
5 Extension of the Fine Topology to P" v { oo I ......Page 204
7. Application to the Fundamental Convergence Theorem and to Reductions. ......Page 206
8. Fine Topology Limits and Euclidean Topology Limits ......Page 207
9. Fine Topology Limits and Euclidean Topology Limits (Continued) ......Page 208
11. Quasi-Lindelof Property ......Page 209
12. Regularity in Terms of the Fine Topology ......Page 210
13. The Euclidean Boundary Set of Thinness of a Greenian Set ......Page 211
15. Characterization ......Page 212
17. The Fine Interior of a Set of Constancy of a Superharmonic Function ......Page 213
18. The Support of a Swept Measure (Continuation of Section 14) ......Page 214
20. A Generalized Reduction ......Page 216
21. Limits of Superharmonic Functions at Irregular Boundary Points of Their Domains ......Page 219
22. The Limit Harmonic Measure fPD ......Page 220
23. Extension of the Domination Principle ......Page 223
1. Motivation ......Page 224
2. The Martin Functions ......Page 225
3. The Martin Space ......Page 226
4. Preliminary Representations of Positive Harmonic Functions and Their Reductions ......Page 228
5. Minimal Harmonic Functions and Their Poles ......Page 229
6. Extension of Lemma 4 ......Page 230
7. The Set of Nonminimal Martin Boundary Points ......Page 231
8. Reductions on the Set of Minimal Martin Boundary Points ......Page 232
9. The Martin Representation ......Page 233
10. Resolutivity of the Martin Boundary ......Page 236
11. Minimal Thinness at a Martin Boundary Point ......Page 237
12. The Minimal-Fine Topology ......Page 239
14. Second Martin Boundary Counterpart of Theorem XI.4(c) ......Page 242
15. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point ......Page 244
17. Minimal-Fine Martin Boundary Limit Functions ......Page 245
18. The Fine Boundary Function of a Potential ......Page 247
19. The Fatou Boundary Limit Theorem for the Martin Space ......Page 248
20. Classical versus Minimal-Fine Topology Boundary Limit Theorems for Relative Superharmonic Functions on a Ball in R" ......Page 250
21. Nontangential and Minimal-Fine Limits at a Half-space Boundary ......Page 251
22. Normal Boundary Limits for a Half-space ......Page 252
23. Boundary Limit Function (Minimal-Fine and Normal) of a Potential on a Half-space ......Page 254
1. Physical Context ......Page 255
2. Measures and Their Energies ......Page 256
3. Charges and Their Energies ......Page 257
4. Inequalities between Potentials, and the Corresponding Energy Inequalities ......Page 258
5. The Function ......Page 259
6. Classical Evaluation of Energy; Hilbert Space Methods ......Page 260
7. The Energy Functional (Relative to an Arbitrary Greenian Subset D of RN) ......Page 262
8. Alternative Proofs of Theorem 7(b') ......Page 264
10. The Classical Capacity Function ......Page 266
11. Inner and Outer Capacities (Notation of Section 10) ......Page 269
12. Extremal Property Characterizations of Equilibrium Potentials (Notation of Section 10) ......Page 270
13. Expressions for C(A) ......Page 272
14. The Gauss Minimum Problems and Their Relation to Reductions ......Page 273
15. Dependence of C' on D ......Page 276
16. Energy Relative to R2 ......Page 277
17. The Wiener Thinness Criterion ......Page 278
18. The Robin Constant and Equilibrium Measures Relative to l 2 (N = 2) ......Page 280
3. Convergence Theorems ......Page 285
5. The Dirichlet Problem (Euclidean Boundary) ......Page 286
6. Green Functions ......Page 287
8. Identification of the Measure Defining a Potential ......Page 288
9. Riesz Decomposition ......Page 289
10. The Martin Boundary ......Page 290
1. Conventions ......Page 291
2. The Parabolic and Coparabolic Operators ......Page 292
3. Coparabolic Polynomials ......Page 293
4. The Parabolic Green Function oflt8. ......Page 295
5. Maximum-Minimum Parabolic Function Theorem ......Page 296
6. Application of Green's Theorem ......Page 298
7. The Parabolic Green Function of a Smooth Domain; The Riesz Decomposition and Parabolic Measure (Formal Treatment) ......Page 299
8. The Green Function of an Interval ......Page 301
9. Parabolic Measure for an Interval ......Page 302
10. Parabolic Averages ......Page 304
11. Harnack's Theorems in the Parabolic Context ......Page 305
12. Superparabolic Functions ......Page 306
13. Superparabolic Function Minimum Theorem ......Page 308
14. The Operation iB and the Defining Average Properties of Superparabolic Functions ......Page 309
15. Superparabolic and Parabolic Functions on a Cylinder ......Page 310
16. The Appall Transformation ......Page 311
17. Extensions of a Parabolic Function Defined on a Cylinder ......Page 312
1. The Parabolic Poisson Integral for a Slab ......Page 314
2. A Generalized Superparabolic Function Inequality ......Page 316
4. A Boundary Limit Criterion for the Identically Vanishing of a Positive Parabolic Function ......Page 317
6. The and D(jie_) Classes of Parabolic Functions on a Slab ......Page 319
7. The Parabolic Boundary Limit Theorem ......Page 321
8. Minimal Parabolic Functions on a Slab ......Page 322
2. The Parabolic Fundamental Convergence Theorem (Preliminary Version) and the Reduction Operation ......Page 324
3. The Parabolic Context Reduction Operations ......Page 325
4. The Parabolic Green Function ......Page 327
5. Potentials ......Page 329
6. The Smoothness of Potentials ......Page 332
8. Parabolic-Polar Sets ......Page 334
9. The Parabolic-Fine Topology ......Page 337
10. Semipolar Sets ......Page 338
11. Preliminary List of Reduction Properties ......Page 339
12. A Criterion of Parabolic Thinness ......Page 342
13. The Parabolic Fundamental Convergence Theorem ......Page 343
14. Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions ......Page 345
16. Parabolic-Reduction Properties ......Page 346
17. Proofs of the Reduction Properties in Section 16 ......Page 349
18. The Classical Context Green Function in Terms of the Parabolic Context Green Function (N > 1) ......Page 355
19. The Quasi-Lindelof Property ......Page 357
1. Relativization of the Parabolic Context; The PWB Method in this Context ......Page 358
2. h-Parabolic Measure ......Page 361
3. Parabolic Barriers ......Page 362
4. Relations between the Classical Dirichlet Problem and the Parabolic Context Dirichlet Problem ......Page 363
5. Classical Reductions in the Parabolic Context ......Page 364
6. Parabolic Regularity of Boundary Points ......Page 366
8. Sweeping in the Parabolic Context ......Page 370
9. The Extension GD of OD and the Parabolic Average to( ,GB when b c h ......Page 372
10. Conditions that E *f ......Page 374
11. Parabolic- and Coparabolic-Polar Sets ......Page 376
12. Parabolic- and Coparabolic-Semipolar Sets ......Page 377
13. The Support of a Swept Measure ......Page 379
14. An Internal Limit Theorem; The Coparabolic-Fine Topology Smoothness of Superparabolic Functions ......Page 380
15. Application to a Version of the Parabolic Context Fatou Boundary Limit Theorem on a Slab ......Page 386
17. Limits of Superparabolic Functions at Parabolic-Irregular Boundary Points of Their Domains ......Page 387
19. Lattices and Related Classes of Functions in the Parabolic Context ......Page 390
1. Introduction ......Page 392
2. The Martin Functions of Martin Point Set and Measure Set Pairs ......Page 393
3. The Martin Space DM ......Page 395
4. Preparatory Material for the Parabolic Context Martin Representation Theorem ......Page 396
5. Minimal Parabolic Functions and Their Poles ......Page 398
6. The Set of Nonminimal Martin Boundary Points ......Page 399
8. Martin Boundary of a Slab D = R' x ]0, b[ with 0 < d S + op ......Page 400
9. Martin Boundaries for the Lower Half-space of A' and for Al ......Page 403
10. The Martin Boundary of ,6 = ]0, +oo[ x ] - oo, 6[ ......Page 404
12. The Minimal-Fine Topology in the Parabolic Context ......Page 406
13. Boundary Counterpart of Theorem XVIII.l4(f) ......Page 408
15. The Parabolic Context Fatou Boundary Limit Theorem on Martin Spaces ......Page 410
Part 2 Probabilistic Counterpart of Part I ......Page 414
1. Adapted Families of Functions on Measurable Spaces ......Page 416
2. Progressive Measurability ......Page 417
3. Random Variables ......Page 419
4. Conditional Expectations ......Page 420
5. Conditional Expectation Continuity Theorem ......Page 422
6. Fatou's Lemma for Conditional Expectations ......Page 425
7. Dominated Convergence Theorem for Conditional Expectations ......Page 426
8. Stochastic Processes, "Evanescent," "Indistinguishable," "Standard Modification," "Nearly ......Page 427
9. The Hitting of Sets and Progressive Measurability ......Page 430
10. Canonical Processes and Finite-Dimensional Distributions ......Page 431
11. Choice of the Basic Probability Space ......Page 433
12. The Hitting of Sets by a Right Continuous Process ......Page 434
13. Measurability versus Progressive Measurability of Stochastic Processes ......Page 436
14. Predictable Families of Functions ......Page 439
1. The Context of Optional Times ......Page 442
2. Optional Time Properties (Continuous Parameter Context) ......Page 444
3. Process Functions at Optional Times ......Page 446
4. Hitting and Entry Times ......Page 448
5. Application to Continuity Properties of Sample Functions ......Page 450
7. Predictable Optional Times ......Page 452
8. Section Theorems ......Page 454
9. The Graph of a Predictable Time and the Entry Time of a Predictable Set ......Page 455
10. Semipolar Subsets of 1R x 0 ......Page 456
11. The Classes D and L?of Stochastic Processes ......Page 457
12. Decomposition of Optional Times; Accessible and Totally Inaccessible Optional Times ......Page 458
1. Definitions ......Page 461
2. Examples ......Page 462
3. Elementary Properties (Arbitrary Simply Ordered Parameter Set) ......Page 464
5. Convergence of Supermartingale Families ......Page 466
6. Optional Sampling Theorem (Bounded Optional Times) ......Page 467
7. Optional Sampling Theorem for Right Closed Processes ......Page 469
9. Maximal Inequalities ......Page 471
11. An L?Inequality for Submartingale Suprema ......Page 473
12. Crossings ......Page 474
13. Forward Convergence in the L' Bounded Case ......Page 479
14. Convergence of a Uniformly Integrable Martingale ......Page 480
15. Forward Convergence of a Right Closable Supermartingale ......Page 482
16. Backward Convergence of a Martingale ......Page 483
18. The r Operator ......Page 484
19. The Natural Order Decomposition Theorem for Supermartingales ......Page 486
20. The Operators LM and GM ......Page 487
22. Potential Theory Reductions in a Discrete Parameter Probability Context ......Page 488
23. Application to the Crossing Inequalities ......Page 490
1. Continuity Properties ......Page 492
2. Optional Sampling of Uniformly Integrable Continuous Parameter Martingales ......Page 497
3. Optional Sampling and Convergence of Continuous Parameter Supermartingales ......Page 499
4. Increasing Sequences of Supermartingales ......Page 502
5. Probability Version of the Fundamental Convergence Theorem of Potential Theory ......Page 505
6. Quasi-Bounded Positive Supermartingales; Generation of Supermartingale Potentials by Increasing Processes ......Page 509
7. Natural versus Predictable Increasing Processes (1= 71' or R+) ......Page 512
8. Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case ......Page 517
9. An Inequality for Predictable Increasing Processes ......Page 518
10. Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets ......Page 519
11 Generation of Supermartingale Potentials by Increasing Processes in the Continuous Parameter Case: The Meyer Decomposition ......Page 522
12. Meyer Decomposition of a Submartingale ......Page 524
13 Role of the Measure Associated with a Supermartingale; The Supermartingale Domination Principle ......Page 525
14. The Operators r, LM, and GM in the Continuous Parameter Context ......Page 529
15. Potential Theory on a8' x 52 ......Page 530
16. The Fine Topology of R * x f) ......Page 531
17 Potential Theory Reductions in a Continuous Parameter Probability Context ......Page 533
18. Reduction Properties ......Page 534
19. Proofs of the Reduction Properties in Section 18 ......Page 538
20. Evaluation of Reductions ......Page 542
21. The Energy of a Supermartingale Potential ......Page 544
22. The Subtraction of a Supermartingale Discontinuity ......Page 545
23. Supermartingale Decompositions and Discontinuities ......Page 547
1. Conventions; The Essential Order ......Page 549
2. LM when ; Is a Submartingale ......Page 550
3. Uniformly Integrable Positive Submartingales ......Page 552
4. L-Bounded Stochastic Processes (p >_ 1) ......Page 553
5. The Lattices ('St, 5), ('S+, 5), (Si, 5), (S+, 5) ......Page 554
6. The Vector Lattices ('S, <) and (S, <) ......Page 557
7. The Vector Lattices ('S.,<) and (S,,,, <) ......Page 558
8. The Vector Lattices ('Si, <) and (Sr, <) ......Page 559
9. The Vector Lattices ('Sqb, <) and (Sqb, <) ......Page 560
10. The Vector Lattices ('Sr, <) and (Sr, <) ......Page 561
11. The Orthogonal Decompositions 'S,,, = 'Smgb + 'S., and S. = S,,,qb + S,,,, ......Page 562
12. Local Martingales and Singular Supermartingale Potentials in (S, <) ......Page 563
13. Quasimartingales (Continuous Parameter Context) ......Page 564
1. The Markov Property ......Page 568
2. Choice of Filtration ......Page 573
3. Integral Parameter Markov Processes with Stationary Transition Probabilities ......Page 574
4. Application of Martingale Theory to Discrete Parameter Markov Processes ......Page 576
5. Continuous Parameter Markov Processes with Stationary Transition Probabilities ......Page 579
6. Specialization to Right Continuous Processes ......Page 581
7. Continuous Parameter Markov Processes: Lifetimes and Trap Points ......Page 583
8. Right Continuity of Markov Process Filtrations; A Zero-One (0-1) Law ......Page 585
9. Strong Markov Property ......Page 586
10. Probabilistic Potential Theory; Excessive Functions ......Page 589
11. Excessive Functions and Supermartingales ......Page 593
12. Excessive Functions and the Hitting Times of Analytic Sets (Notation and Hypotheses of Section 11) ......Page 594
13. Conditioned Markov Processes ......Page 595
14. Tied Down Markov Processes ......Page 596
15. Killed Markov Processes ......Page 597
1. Processes with Independent Increments and State Space I8" ......Page 599
2. Brownian Motion ......Page 601
3. Continuity of Brownian Paths ......Page 605
4. Brownian Motion Filtrations ......Page 607
5. Elementary Properties of the Brownian Transition Density and Brownian Motion ......Page 610
6. The Zero-One Law for Brownian Motion ......Page 612
7. Tied Down Brownian Motion ......Page 615
8. Andre Reflection Principle ......Page 616
9. Brownian Motion in an Open Set (N z I) ......Page 618
10. Space-Time Brownian Motion in an Open Set ......Page 621
11. Brownian Motion in an Interval ......Page 623
12. Probabilistic Evaluation of Parabolic Measure for an Interval ......Page 624
13. Probabilistic Significance of the Heat Equation and Its Dual ......Page 625
1. Notation ......Page 628
2. The Size of r. ......Page 630
3. Properties of the Ito Integral ......Page 631
4. The Stochastic Integral for an Integrand Process in ro ......Page 634
5. The Stochastic Integral for an Integrand Process in r ......Page 635
6. Proofs of the Properties in Section 3 ......Page 636
7. Extension to Vector-Valued and Complex-Valued Integrands ......Page 640
8 Martingales Relative to Brownian Motion Filtrations ......Page 641
9. A Change of Variables ......Page 644
10. The Role of Brownian Motion Increments ......Page 647
11 (N = I) Computation of the Ito Integral by Riemann-Stieltjes Sums ......Page 649
12. Ito's Lemma ......Page 650
13. The Composition of the Basic Functions of Potential Theory with Brownian Motion ......Page 654
14. The Composition of an Analytic Function with Brownian Motion ......Page 655
1 Elementary Martingale Applications ......Page 656
2. Coparabolic Polynomials and Martingale Theory ......Page 659
3. Superharmonic and Harmonic Functions on R' and Supermartingales and Martingales ......Page 661
4 Hitting of an f. Set ......Page 664
5. The Hitting of a Set by Brownian Motion ......Page 665
6 Superharmonic Functions, Excessive for Brownian Motion ......Page 666
7 Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion; A Probabilistic Fatou Boundary Limit Theorem. ......Page 670
8 Excessive and Invariant Functions for Brownian Motion ......Page 674
9. Application to Hitting Probabilities and to Parabolicity of Transition Densities. ......Page 676
10. (N = 2) The Hitting of Nonpolar Sets by Brownian Motion ......Page 677
11. Continuity of the Composition of a Function with Brownian Motion ......Page 678
12. Continuity of Superharmonic Functions on Brownian Motion ......Page 679
13 Preliminary Probabilistic Solution of the Classical Dirichlet Problem ......Page 680
14 Probabilistic Evaluation of Reductions ......Page 682
15 Probabilistic Description of the Fine Topology ......Page 685
16 x-Excessive Functions for Brownian Motion and Their Composition with Brownian Motions ......Page 688
17. Brownian Motion Transition Functions as Green Functions; The Corresponding Backward and Forward Parabolic Equations ......Page 690
18. Excessive Measures for Brownian Motion ......Page 692
20. Brownian Motion into a Set from an Irregular Boundary Point ......Page 695
1. Definition ......Page 697
2. h-Brownian Motion in Terms of Brownian Motion ......Page 700
3. Contexts for (2. 1) ......Page 705
4. Asymptotic Character of h-Brownian Paths at Their Lifetimes ......Page 706
5. h-Brownian Motion from an Infinity of h ......Page 709
6. Brownian Motion under Time Reversal ......Page 711
7. Preliminary Probabilistic Solution of the Dirichlet Problem for h-Harmonic Functions; h-Brownian Motion Hitting Probabilities and the Corresponding Generalized Reductions ......Page 713
8. Probabilistic Boundary Limit and Internal Limit Theorems for Ratios of Strictly Positive Superharmonic Functions ......Page 717
9. Conditional Brownian Motion in a Ball ......Page 720
10. Conditional Brownian Motion Last Hitting Distributions; The Capacitary Distribution of a Set in Terms of a Last Hitting Distribution ......Page 722
11. The Tail a Algebra of a Conditional Brownian Motion ......Page 723
12. Conditional Space-Time Brownian Motion ......Page 728
13. [Space-Time] Brownian Motion in [iv] ii1;N with Parameter Set R ......Page 729
Part 3 ......Page 732
1. Correspondence between Classical Potential Theory and Martingale Theory ......Page 734
3. The Classes L?and D ......Page 735
4. PWB-Related Conditions on h-Harmonic Functions and on Martingales ......Page 736
5. Class D Property versus Quasi-Boundedness ......Page 737
6. A Condition for Quasi-Boundedness ......Page 738
7. Singularity of an Element of S : ......Page 739
8. The Singular Component of an Element of S4 ......Page 740
9. The Class S ......Page 741
10. The Class S. ......Page 743
11. Lattice Theoretic Analysis of the Composition of an h-Superharmonic Function with an h-Brownian Motion ......Page 744
12. A Decomposition of Sm, (Potential Theory Context) ......Page 745
13. Continuation of Section I l ......Page 746
1. Context of the Problem ......Page 748
2. Probabilistic Analysis of the PWB Method ......Page 749
3 PW B' Examples ......Page 752
4. Tail a Algebras in the PWB'' Context ......Page 754
1. The Structure of Brownian Motion on the Martin Space ......Page 756
2. Brownian Motions from Martin Boundary Points (Notation of Section I) ......Page 757
3. The Zero-One Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the Minimal-Fine Topology (Notation of Section I) ......Page 759
4. The Probabilistic Fatou Theorem on the Martin Space ......Page 761
5. Probabilistic Approach to Theorem I.XI.4(c) and Its Boundary Counterparts ......Page 762
6. Martin Representation of Harmonic Functions in the Parabolic Context ......Page 764
Appendixes ......Page 768
2. Suslin Schemes ......Page 770
3 Sets Analytic over a Product Paving ......Page 771
5 Projection Characterization %f (V) ......Page 772
7. Projections of Sets in Product Pavings ......Page 773
9. The G, Sets of a Complete Metric Space ......Page 774
11 The Baire Null Space ......Page 775
12. Analytic Sets ......Page 776
13. Analytic Subsets of Polish Spaces ......Page 777
2. Sierpinski Lemma ......Page 779
4. Lusin's Theorem ......Page 780
6. Strongly Subadditive Set Functions ......Page 781
7. Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function ......Page 782
8. Topological Precapacities ......Page 784
9. Universally Measurable Sets ......Page 785
3. Cones ......Page 787
4. The Specific Order Generated by a Cone ......Page 788
5. Vector Lattices ......Page 789
8. Bands in a Vector Lattice ......Page 791
9. Projections on Bands ......Page 792
11. The Band Generated by a Single Element ......Page 793
12. Order Convergence ......Page 794
13. Order Convergence on a Linearly Ordered Set ......Page 795
2. Measurable Spaces and Measurable Functions ......Page 796
3. Composition of Functions ......Page 797
4. The Measure Lattice of a Measurable Space ......Page 798
5. The a Finite Measure Lattice of a Measurable Space (Notation of Section 4) ......Page 800
7. The Vector Lattice 4l ......Page 801
8. Absolute Continuity and Singularity ......Page 802
9. Lattices of Measurable Functions on a Measure Space ......Page 803
10. Order Convergence of Families of Measurable Functions ......Page 804
11. Measures on Polish Spaces ......Page 806
12. Derivates of Measures ......Page 807
Appendix V Uniform Integrability ......Page 808
1. Kernels ......Page 810
3. Transition Functions ......Page 811
1. An Elementary Limit Theorem ......Page 814
3. A One-Dimensional Ratio Integral Limit Theorem ......Page 815
4. A Ratio Integral Limit Theorem Involving Convex Variational Derivates. ......Page 817
2. Suprema of Families of Lower Semicontinuous Functions. ......Page 820
3 Choquet Topological Lemma ......Page 821
Part 1 ......Page 822
Part 2 ......Page 835
Part 3 ......Page 844
Appendixes ......Page 845
Bibliography ......Page 848
Notation Index ......Page 856
Index ......Page 858
Back Cover......Page 876