扩散过程及其样本轨道 英文版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

扩散过程及其样本轨道 英文版PDF格式电子书版下载

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Prerequisites1

Chapter 1．The standard BROWNian motion5

1.1．The standard random walk5

1.2．Passage times for the standard random walk7

1.3．HIN？IN'S proof of the DE MOIVRE-LAPLACE limit theorem10

1.4．The standard BROWNian motion12

1.5．P.L？VY'S construction19

1.6．Strict MARKOV character22

1.7．Passage times for the standard BROWNian motion25

Note 1:Homogeneous differential processes with increasing paths31

1.8．KOLMOGOROV'S test and the law of the iterated logarithm33

1.9．P.L？VY'S H？LDER condition36

1.10．Approximating the BROWNian motion by a random walk38

Chapter 2．BROWNian local times40

2.1．The reflecting BROWNian motion40

2.2．P.L？VY'S local time42

2.3．Elastic BROWNian motion45

2.4．t+ and down-crossings48

2.5．t+ as HAUSDORFF-BESICOVITCH 1/2-dimensional measure50

Note 1:Submartingales52

Note 2:HAUSDORFF measure and dimension53

2.6．KAC'S formula for BROWNian functionals54

2.7．BESSEL processes59

2.8．Standard BROWNian local time63

2.9．BROWNian excursions75

2.10．Application of the BESSEL process to BROWNian excursions79

2.11．A time substitution81

Chapter 3．The general 1-dimensional diffusion83

3.1．Definition83

3.2．MARKOV times86

3.3．Matching numbers89

3.4．Singular points91

3.5．Decomposing the general diffusion into simple pieces92

3.6．GREEN operators and the space D94

3.7．Generators98

3.8．Generators continued100

3.9．Stopped diffusion102

Chapter 4．Generators105

4.1．A general view105

4.2．？ as local differential operator:conservative non-singular case111

4.3．？ as local differential operator:general non-singular case116

4.4．A second proof119

4.5．？ at an isolated singular point125

4.6．Solving ？·u=αu128

4.7．？ as global differential operator:non-singular case135

4.8．？ on the shunts136

4.9．？ as global differential operator:singular case142

4.10．Passage times144

Note 1:Differential processes with increasing paths146

4.11．Eigen-differential expansions for GREEN functions and transition densities149

4.12．KOLMOGOROV'S test161

Chapter 5．Time changes and killing164

5.1．Construction of sample paths:a general view164

5.2．Time changes:Q=R1167

5.3．Time changes:Q=[O,+∞)171

5.4．Local times174

5.5．Subordination and chain rule176

5.6．Killing times179

5.7．FELLER'S BROWNian motions186

5.8．IKEDA'S example188

5.9．Time substitutions must come from local time integrals190

5.10．Shunts191

5.11．Shunts with killing196

5.12．Creation of mass200

5.13．A parabolic equation201

5.14．Explosions206

5.15．A non-linear parabolic equation209

Chapter 6．Local and inverse local times212

6.1．Local and inverse local times212

6.2．L？VY measures214

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 downcrossings222

6.5b t as HAUSDORFF measure223

6.5c t as diffusion223

6.5d Excursions223

6.6．Dimension numbers224

6.7．Comparison tests225

Note 1:Dimension numbers and fractional dimensional capacities227

6.8．An individual ergodic theorem228

Chapter 7．BROWNian motion in several dimensions232

7.1．Diffusion in several dimensions232

7.2．The standard BROWNian motion in several dimensions233

7.3．Wandering out to ∞236

7.4．GREENian domains and GREEN functions237

7.5．Excessive functions243

7.6．Application to the spectrum of ？/2245

7.7．Potentials and hitting probabilities247

7.8．NEWTONian capacities250

7.9．GAUSS'S quadratic form253

7.10．WIENER'S test255

7.11．Applications of WIENER'S test257

7.12．DIRICHLET problem261

7.13．NEUMANN problem264

7.14．Space-time BROWNian motion266

7.15．Spherical BROWNian motion and skew products269

7.16．Spinning274

7.17．An individual ergodic theorem for the standard 2-dimensional BROWNian motion277

7.18．Covering BROWNian motions279

7.19．Diffusions with BROWNian hitting probabilities283

7.20．Right-continuous paths286

7.21．RIESZ potentials288

Chapter 8．A general view of diffusion in several dimensions291

8.1．Similar diffusions291

8.2．？ as differential operator293

8.3．Time substitutions295

8.4．Potentials296

8.5．Boundaries299

8.6．Elliptic operators302

8.7．FELLER'S little boundary and tail algebras303

Bibliography306

List of notations313

Index315