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