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