Prerequisites 1
Chapter 1．The standard BROWNian motion 5
1.1．The standard random walk 5
1.2．Passage times for the standard random walk 7
1.3．HIN？IN'S proof of the DE MOIVRE-LAPLACE limit theorem 10
1.4．The standard BROWNian motion 12
1.5．P.L？VY'S construction 19
1.6．Strict MARKOV character 22
1.7．Passage times for the standard BROWNian motion 25
Note 1:Homogeneous differential processes with increasing paths 31
1.8．KOLMOGOROV'S test and the law of the iterated logarithm 33
1.9．P.L？VY'S H？LDER condition 36
1.10．Approximating the BROWNian motion by a random walk 38
Chapter 2．BROWNian local times 40
2.1．The reflecting BROWNian motion 40
2.2．P.L？VY'S local time 42
2.3．Elastic BROWNian motion 45
2.4．t+ and down-crossings 48
2.5．t+ as HAUSDORFF-BESICOVITCH 1/2-dimensional measure 50
Note 1:Submartingales 52
Note 2:HAUSDORFF measure and dimension 53
2.6．KAC'S formula for BROWNian functionals 54
2.7．BESSEL processes 59
2.8．Standard BROWNian local time 63
2.9．BROWNian excursions 75
2.10．Application of the BESSEL process to BROWNian excursions 79
2.11．A time substitution 81
Chapter 3．The general 1-dimensional diffusion 83
3.1．Definition 83
3.2．MARKOV times 86
3.3．Matching numbers 89
3.4．Singular points 91
3.5．Decomposing the general diffusion into simple pieces 92
3.6．GREEN operators and the space D 94
3.7．Generators 98
3.8．Generators continued 100
3.9．Stopped diffusion 102
Chapter 4．Generators 105
4.1．A general view 105
4.2．？ as local differential operator:conservative non-singular case 111
4.3．？ as local differential operator:general non-singular case 116
4.4．A second proof 119
4.5．？ at an isolated singular point 125
4.6．Solving ？·u=αu 128
4.7．？ as global differential operator:non-singular case 135
4.8．？ on the shunts 136
4.9．？ as global differential operator:singular case 142
4.10．Passage times 144
Note 1:Differential processes with increasing paths 146
4.11．Eigen-differential expansions for GREEN functions and transition densities 149
4.12．KOLMOGOROV'S test 161
Chapter 5．Time changes and killing 164
5.1．Construction of sample paths:a general view 164
5.2．Time changes:Q=R1 167
5.3．Time changes:Q=[O,+∞) 171
5.4．Local times 174
5.5．Subordination and chain rule 176
5.6．Killing times 179
5.7．FELLER'S BROWNian motions 186
5.8．IKEDA'S example 188
5.9．Time substitutions must come from local time integrals 190
5.10．Shunts 191
5.11．Shunts with killing 196
5.12．Creation of mass 200
5.13．A parabolic equation 201
5.14．Explosions 206
5.15．A non-linear parabolic equation 209
Chapter 6．Local and inverse local times 212
6.1．Local and inverse local times 212
6.2．L？VY measures 214
6.3．t and the intervals of [O,+∞)-？ 218
6.4．A counter example:t and the intervals of [O,+∞)-？ 220
6.5a t and downcrossings 222
6.5b t as HAUSDORFF measure 223
6.5c t as diffusion 223
6.5d Excursions 223
6.6．Dimension numbers 224
6.7．Comparison tests 225
Note 1:Dimension numbers and fractional dimensional capacities 227
6.8．An individual ergodic theorem 228
Chapter 7．BROWNian motion in several dimensions 232
7.1．Diffusion in several dimensions 232
7.2．The standard BROWNian motion in several dimensions 233
7.3．Wandering out to ∞ 236
7.4．GREENian domains and GREEN functions 237
7.5．Excessive functions 243
7.6．Application to the spectrum of ？/2 245
7.7．Potentials and hitting probabilities 247
7.8．NEWTONian capacities 250
7.9．GAUSS'S quadratic form 253
7.10．WIENER'S test 255
7.11．Applications of WIENER'S test 257
7.12．DIRICHLET problem 261
7.13．NEUMANN problem 264
7.14．Space-time BROWNian motion 266
7.15．Spherical BROWNian motion and skew products 269
7.16．Spinning 274
7.17．An individual ergodic theorem for the standard 2-dimensional BROWNian motion 277
7.18．Covering BROWNian motions 279
7.19．Diffusions with BROWNian hitting probabilities 283
7.20．Right-continuous paths 286
7.21．RIESZ potentials 288
Chapter 8．A general view of diffusion in several dimensions 291
8.1．Similar diffusions 291
8.2．？ as differential operator 293
8.3．Time substitutions 295
8.4．Potentials 296
8.5．Boundaries 299
8.6．Elliptic operators 302
8.7．FELLER'S little boundary and tail algebras 303
Bibliography 306
List of notations 313
Index 315