Probability and Stochastic Processes for Physicists

Preface......Page 7
Contents......Page 9
Part I Probability......Page 14
1.1 Samples......Page 15
1.2 Events......Page 18
1.3 Probability......Page 23
1.4 Conditional Probability......Page 26
1.5 Independent Events......Page 29
Reference......Page 30
2.1.1 Finite and Countable Spaces......Page 31
2.1.2 Bernoulli Trials......Page 33
2.2.1 Cumulative Distribution Functions......Page 38
2.2.2 Discrete Distributions......Page 40
2.2.3 Absolutely Continuous Distributions: Density......Page 42
2.2.5 Mixtures......Page 46
2.3.1 Multivariate Distribution Functions......Page 48
2.3.2 Multivariate Densities......Page 50
2.3.3 Marginal Distributions......Page 52
2.3.4 Copulas......Page 55
2.4 Distributions on Rinfty and RT......Page 57
References......Page 60
3.1.1 Measurability......Page 61
3.1.2 Laws and Distributions......Page 63
3.1.3 Generating New rv\'s......Page 65
3.2.1 Random Elements......Page 67
3.2.2 Joint and Marginal Distributions and Densities......Page 69
3.2.3 Independence of rv\'s......Page 73
3.2.4 Decomposition of Binomial rv\'s......Page 75
3.3.1 Integration and Expectation......Page 78
3.3.2 Change of Variables......Page 81
3.3.3 Variance and Covariance......Page 87
3.4.1 Conditional Distributions......Page 93
3.4.2 Conditional Expectation......Page 96
3.4.3 Optimal Mean Square Estimation......Page 102
3.5.1 Functions of rv\'s......Page 104
3.5.2 Sums of Independent rv\'s......Page 107
References......Page 110
4.1 Convergence......Page 111
4.2.1 Definition and Properties......Page 114
4.2.2 Gaussian Laws......Page 120
4.2.3 Composition and Decomposition of Laws......Page 123
4.3 Laws of Large Numbers......Page 125
4.4 Gaussian Theorems......Page 129
4.5 Poisson Theorems......Page 131
4.6 Where the Classical Limit Theorems Fail......Page 136
References......Page 138
Part II Stochastic Processes......Page 139
5.1 Identification and Distribution......Page 140
5.2 Expectations and Correlations......Page 143
5.3 Convergence and Continuity......Page 144
5.4 Differentiation and Integration in ms......Page 145
5.5 Stationarity and Ergodicity......Page 148
5.6 Power Spectrum......Page 152
References......Page 154
6.1.1 Point Processes and Renewals......Page 155
6.1.2 Poisson Process......Page 160
6.1.3 Compensated Poisson Process......Page 167
6.1.4 Compound Poisson Process......Page 168
6.1.5 Shot Noise......Page 171
6.2.1 Random Walk......Page 174
6.2.2 Wiener Process......Page 175
6.2.3 Geometric Wiener Process......Page 181
6.3 White Noise......Page 183
6.4 Brownian Motion......Page 186
6.4.1 Einstein (1905)......Page 187
6.4.2 Langevin (1908)......Page 190
References......Page 191
7.1.1 Markov Property......Page 193
7.1.2 Chapman-Kolmogorov Equations......Page 197
7.1.3 Independent Increments Processes......Page 200
7.1.4 Stationarity and Homogeneity......Page 202
7.1.5 Distribution Ergodicity......Page 204
7.1.6 Lévy Processes......Page 207
7.1.7 Continuity and Jumps......Page 208
7.1.8 Poisson, Wiener and Cauchy Processes......Page 209
7.1.9 Ornstein-Uhlenbeck Processes......Page 214
7.1.10 Non Markovian, Gaussian Processes......Page 216
7.2 Jump-Diffusion Processes......Page 218
7.2.1 Forward Equations......Page 219
7.2.2 Backward Equations......Page 223
7.2.3 Main Classes of Jump-Diffusions......Page 225
7.2.4 Notable Jump-Diffusion Processes......Page 229
References......Page 233
8.1 Wienerian White Noise......Page 234
8.2 Stochastic Integration......Page 237
8.2.1 Wiener Integral......Page 238
8.2.2 Itō Integral......Page 240
8.3.1 Elementary Integration Rules......Page 242
8.3.2 Expectation and Covariance......Page 246
8.3.3 Stochastic Infinitesimals......Page 247
8.3.4 Differentiation Rules......Page 250
8.4.1 Stochastic Differentials and Itō Formula......Page 253
8.4.2 The SDE\'s and Their Solutions......Page 254
8.4.3 SDE\'s and Fokker-Planck Equations......Page 256
8.5 Notable SDE\'s......Page 257
8.5.2 SDE\'s with Time Dependent Coefficients......Page 258
8.5.3 SDE\'s with No Drift and x-Linear Diffusion......Page 260
8.5.4 SDE\'s with x-Linear Drift and Constant Diffusion......Page 263
References......Page 265
Part III Physical Modeling......Page 266
9.1 Free Brownian Particle......Page 267
9.2 Ornstein-Uhlenbeck Versus Einstein-Smoluchowski......Page 270
9.3 Ornstein-Uhlenbeck Markovianity......Page 272
9.4 Brownian Particle in a Force Field......Page 275
9.5 Boltzmann Distribution......Page 278
References......Page 282
10.1 Madelung Decomposition......Page 284
10.2 Retarded and Advanced Equations......Page 286
10.3.1 Forward and Backward Mean Derivatives......Page 291
10.3.2 Mean Velocities......Page 293
10.3.3 Mean Acceleration......Page 296
10.4 Dynamics of a Diffusion Process......Page 299
10.4.1 Time Reversal Invariance......Page 303
References......Page 306
Appendix A Consistency (Sect. 2.3.4摥映數爠eflinkcopule2.3.42)......Page 307
Appendix B Inequalities (Sect. 3.3.2摥映數爠eflinkcambvar3.3.23)......Page 315
Appendix C Bertrand\'s Paradox (Sect. 3.5.1摥映數爠eflinkfunzval3.5.13)......Page 318
Appendix D Lp Spaces of rv\'s (Sect. 4.1摥映數爠eflinkconvsucc4.14)......Page 322
Appendix E Moments and Cumulants (Sect. 4.2.1摥映數爠eflinkproprfc4.2.14)......Page 324
Appendix F Binomial Limit Theorems (Sect. 4.3摥映數爠eflinkteorgn4.34)......Page 327
Appendix G Non Uniform Point Processes (Sect. 6.1.1摥映數爠eflinkpuntirinnovi6.1.16)......Page 331
Appendix H Stochastic Calculus Paradoxes (Sect. 6.4.2摥映數爠eflinklangev6.4.26)......Page 333
Appendix I Pseudo-Markovian Processes (Sect. 7.1.2摥映數爠eflinkeqchk7.1.27)......Page 338
Appendix J Fractional Brownian Motion (Sect. 7.1.10摥映數爠eflinkgaussproc7.1.107)......Page 341
Appendix K Ornstein-Uhlenbeck Equations (Sect. 7.2.4摥映數爠eflinkeqparticolari7.2.47)......Page 343
Appendix L Stratonovich Integral (Sect. 8.2.2摥映數爠eflinkintito8.2.28)......Page 346
Appendix M Stochastic Bridges (Sect. 10.2摥映數爠eflinkadvret10.210)......Page 348
Appendix N Kinematics of Gaussian Diffusions (Sect. 10.3.1摥映數爠eflinkmeanderiv110.3.110)......Page 352
Appendix O Substantial Operators (Sect. 10.3.3摥映數爠eflinkstreams10.3.310)......Page 363
Appendix P Constant Diffusion Coefficients (Sect. 10.4摥映數爠eflinkschroed10.410)......Page 365
BookmarkTitle:......Page 367
Index......Page 368