扩散 马尔可夫过程和鞅 第1卷PDF电子书下载

CHAPTER Ⅰ．BROWNIAN MOTION 1

1．INTRODUCTION 1

1 What is Brownian motion,and why study it? 1

2．Brownian motion as a martingale 2

3．Brownian motion as a Gaussian process 3

4．Brownian motion as a Markov process 5

5 Brownian motion as a diffusion（and martingale） 7

2．BASICS ABOUT BROWNIAN MOTION 10

6 Existence and uniqueness of Brownian motion 10

7．Skorokhod embedding 13

8．Donsker's Invariance Principle 16

9．Exponential martingales and first-passage distributions 18

10．Some sample-path properties 19

11．Quadratic variation 21

12．The strong Markov property 21

13．Reflection 25

14．Reflecting Brownian motion and local time 27

15．Kolmogorov's test 31

16．Brownian exponential martingales and the Law of the Iterated Logarithm 31

3．BROWNIAN MOTION IN HIGHER DIMENSIONS 36

17．Some martingales for Brownian motion 36

18．Recurrence and transience in higher dimensions 38

19．Some applications of Brownian motion to complex analysis 39

20．Windings of planar Brownian motion 43

21．Multiple points,cone points,cut points 45

22．Potential theory of Brownian motion in IRd（d≥3） 46

23．Brownian motion and physical diffusion 51

4．GAUSSIAN PROCESSES AND LEVY PROCESSES 55

Gaussian processes 55

24．Existence results for Gaussian processes 55

25 Continuity results 59

26．Isotropic random flows 66

27．Dynkin's Isomorphism Theorem 71

Lévy processes 73

28．Lévy processes 73

29．Fluctuation theory and Wiener-Hopf factorisation 80

30．Local time of Levy processes 82

CHAPTER Ⅱ．SOME CLASSICAL THEORY 85

1．BASIC MEASURE THEORY 85

Measurability and measure 85

1．Measurable spaces;σ-algebras;π-systems;d-systems 85

2．Measurable functions 88

3．Monotone-Class Theorems 90

4．Measures;the uniqueness lemma;almost everywhere;a.e．（μ，∑） 91

5．Carathéodory's Extension Theorem 93

6．Inner and outerμ-measures;completion 94

Integration 95

7．Definition of the integral∫fdμ 95

8．Convergence theorems 96

9．The Radon-Nikodym Theorem;absolute continuity;λ《μnotation;equivalent measures 98

10．Inequalities;Lp and Lp spaces（p≥1） 99

Product structures 101

11．Productσ-algebras 101

12．Product measure;Fubini's Theorem 102

13．Exercises 104

2．BASIC PROBABILITY THEORY 108

Probability and expectation 108

14．Probability triple;almost surely （a.s.）;a.s.（P），a.s.（P，F） 108

15．lim sup En;First Borel-Cantelli Lemma 109

16．Law of random variable;distribution function;joint law 110

17．Expectation;E（X;F） 110

18．Inequalities：Markov,Jensen,Schwarz,Tchebychev 111

19．Modes of convergence of random variables 113

Uniform integrability and L1 convergence 114

20．Uniform integrability 114

21．L1 convergence 115

Independence 116

22．Independence ofσ-algebras and of random variables 116

23 Existence of families of independent variables 118

24．Exercises 119

3．STOCHASTIC PROCESSES 119

The Daniell-Kolmogorov Theorem 119

25．（ET，ET）;σ-algebras on function space;cylinders andσ-cylinders 119

26．Infinite products of probability triples 121

27．Stochastic process;sample function;law 121

28．Canonical process 122

29．Finite-dimensional distributions;sufficiency;compatibility 123

30．The Daniell-Kolmogorov（DK）Theorem：compact metrizable'case 124

31．The Daniell-Kolmogorov（DK）Theorem：general case 126

32．Gaussian processes;pre-Brownian motion 127

33．Pre-Poisson set functions 128

Beyond the DK Theorem 128

34．Limitations of the DK Theorem 128

35．The role of outer measures 129

36．Modifications;indistinguishability 130

37．Direct const ruction of Poisson measures and subordinators,and of local time from the zero set;Azéma's martingale 131

38．Exercises 136

4．DISCRETE-PARAMETER MARTINGALE THEORY 137

Conditional expectation 137

30．Fundamental theorem and definition 137

40．Notation;agreement with elementary usage 138

41．Properties of conditional expectation：a list 139

42．The role of versions;regular conditional probabilities and pdfs 140

43．A counterexample 141

44．A uniform-integrability property of conditional expectations （Discrete-parameter）martingales and supermartingales 142

45．Filtration;filtered space;adapted process;natural filtration 143

46 Martingale;supermartingale;submartingale 144

47 Previsible process;gambling strategy;a fundamental principle 144

48．Doob's Upcrossing Lemma 145

49．Doob's Supermartingale-Convergence Theorem 146

50．L1 convergence and the UI property 147

51．The Lévy-Doob Downward Theorem 148

52．Doob's Submartingale and Lp Inequalities 150

53．Martingales in L2：orthogonality of increments 152

54．Doob decomposition 153

55．The〈M〉and［M］processes 154

Stopping times,optional stopping and optional sampling 155

56．Stopping time 155

57．Optional-stopping theorems 156

58．The pre-Tσ-algebraFT 158

59．Optional sampling 159

60．Exercises 161

5．CONTINUOUS-PARAMETER SUPERMARTINGALES 163

Regularisation：R-supermartingales 163

61．Orientation 163

62．Some real-variable results 163

63．Filtrations;supermartingales;R-processes,R-supermartingales 166

64．Some important examples 167

65．Doob's Regularity Theorem：Part 1 169

66．Partial augmentation 171

67．Usual conditions;R-filtered space;usual augmentation;R-regularisation 172

68．A necessary pause for thought 174

69．Convergence theorems for R-supermartingales 175

70．Inequalities and Lp convergence for R-submartingales 177

71．Martingale proof of Wiener's Theorem;canonical Brownian motion 178

72．Brownian motion relative to a filtered space 180

Stopping times 181

73．Stopping time T;pre-Tσ-algebra GT;progressive process 181

74．First-entrance（début）times;hitting times;first-approach times：the easy cases 183

75．Why‘completion'in the usual conditions has to be introduced 184

76 Début and Section Theorems 186

77．Optional Sampling for R-supermartingales under the usual conditions 188

78 Two important results for Markov-process theory 191

79．Exercises 192

6．PROBABI LITY M EASU RES ON LUSIN SPACES 200

‘Weak convergence' 202

80 C（J）and Pr（J）when J is compact Hausdorff 202

81．C（J）and Pr（J）when J is compact metrizable 203

82．Polish and Lusin spaces 205

83．The Cb（S）topology of Pr（S）when S is a Lusin space;Prohorov's Theorem 207

84．Some useful convergence results 211

85．Tightness in Pr（W）when W is the path-space W：=C（［0，∞）;IR） 213

86．The Skorokhod representation of Ch（S）convergence on Pr（S） 215

87．Weak convergence versus convergence of finite-dimensional distributions 216

Regular conditional probabilities 217

88．Some preliminaries 217

89．The main existence theorem 218

90．Canonical Brownian Motion CBM（IRN）;Markov property of Px laws 220

91．Exercises 222

CHAPTER Ⅲ．MARKOV PROCESSES 227

1．TRANSITION FUNCTIONS AND RESOLVENTS 227

1．What is a（continuous-time）Markov process? 227

2．The finite-state-space Markov chain 228

3．Transition functions and their resolvents 231

4．Contraction semigroups on Banach spaces 234

5．The Hille-Yosida Theorem 237

2．FELLER-DYNKIN PROCESSES 240

6．Feller-Dynkin（FD）semigroups 240

7．The existence theorem：canonical FD processes 243

8．Strong Markov property：preliminary version 247

9．Strong Markov property：full version;Blumenthal's 0-1 Law 249

10．Some fundamental martingales；Dynkin's formula 252

11．Quasi-left-continuity 255

12．Characteristic operator 256

13．Feller-Dynkin diffusions 258

14．Characterisation of continuous real Lévy processes 261

15．Consolidation 262

3．ADDITIVE FUNCTIONALS 263

16．PCHAFs；λ-excessive functions；Brownian local time 263

17．Proof of the Volkonskii-？ur-Meyer Theorem 267

18．Killing 269

19．The Feynmann-Kac formula 272

20．A Ciesielski-Taylor Theorem 275

21．Time-substitution 277

22 Reflecting Brownian motion 278

23．The Feller-McKean chain 281

24．Elastic Brownian motion；the arcsine law 282

4．APPROACH TO RAY PROCESSES：THE MARTIN BOUNDARY 284

25．Ray processes and Markov chains 284

26．Important example：birth process 286

27．Excessive functions,the Martin kernel and Choquet theory 288

28．The Martin compactification 292

29．The Martin representation;Doob-Hunt explanation 295

30．R.S.Martin's boundary 297

31．Doob-Hunt theory for Brownian motion 298

32．Ray processes and right processes 302

5．RAY PROCESSES 303

33．Orientation 303

34．Ray resolvents 304

35．The Ray-Knight compactification 306

Ray's Theorem：analytical part 309

36．From semigroup to resolvent 309

37．Branch-points 313

38．Choquet rep resentation of l-excessive probability measures 315

Ray's Theorem：probabilistic part 316

39．The Ray process associated with a given entrance law 316

40．Strong Markov property of Ray processes 318

41．The role of branch-points 319

6．APPLICATIONS 321

Martin boundary theory in retrospect 321

42．From discrete to continuous time 321

43．Proof of the Doob-Hunt Convergence Theorem 323

44．The Choquet representation of ∏-excessive functions 325

45．Doob h-transforms 327

Time reversal and related topics 328

46．Nagasawa's formula for chains 328

47．Strong Markov property under time reversal 330

48．Equilibrium charge 331

49．BM（IR）and BES（3）：splitting times 332

A first look at Markov-chain theory 334

50．Chains as Ray processes 334

51．Significance of qi 337

52．Taboo probabilities;first-entrance decomposition 337

53．The Q-matrix;DK conditions 339

54．Local-character condition for Q 340

55．Totally instantaneous Q-matrices 342

56．Last exits 343

57．Excursions from b 345

58．Kingman's solution of the‘Markov characterization problem' 347

59．Symmetrisable chains 348

60．An open problem 349

References for Volumes 1 and 2 351

Index to Volumes 1 and 2 375