1 The Physical Brownian Motion: Diffusion And Noise ............ 1
1.1 Einstein's theory of diffusion .......................... 1
1.2 The velocity process and Langevin's approach ............ 5
1.3 The displacement process ............................... 10
1.4 Classical theory of noise .............................. 13
1.5 An application: Johnson noise .......................... 16
1.6 Linear systems ......................................... 21
2 The Probability Space of Brownian Motion .................... 25
2.1 Introduction ........................................... 25
2.2 The space of Brownian trajectories ..................... 27
2.2.1 The Wiener measure of Brownian trajectories ..... 37
2.2.2 The MBM in  d ................................... 44
2.3 Constructions of the MBM ............................... 46
2.3.1 The Paley-Wiener construction of the Brownian
motion .......................................... 46
2.3.2 P. Lévy's method and refinements ................ 49
2.4 Analytical and statistical properties of Brownian
paths .................................................. 52
2.4.1 The Markov property of the MBM .................. 55
2.4.2 Reflecting and absorbing walls .................. 56
2.4.3 MBM and martingales ............................. 60
3 ltd Integration and Calculus ................................ 63
3.1 Integration of white noise ............................. 63
3.2 The Itô, Stratonovich, and other integrals ............. 66
3.2.1 The Itô integral ................................ 66
3.2.2 The Stratonovich integral ....................... 68
3.2.3 The backward integral ........................... 73
3.3 The construction of the Itô integral ................... 74
3.4 The Itô calculus ....................................... 81
4 Stochastic Differential Equations ........................... 92
4.1 Itô and Stratonovich SDEs .............................. 93
4.2 Transformations of Itô equations ....................... 97
4.3 Solutions of SDEs are Markovian ....................... 101
4.4 Stochastic and partial differential equations ......... 104
4.4.1 The Andronov-Vitt-Pontryagin equation .......... 109
4.4.2 The exit distribution .......................... 1ll
4.4.3 The PDF of the FPT ............................. 114
4.5 The Fokker-Planck equation ............................ 119
4.6 The backward Kolmogorov equation ...................... 124
4.7 Appendix: Proof of Theorem 4.1.1 ...................... 125
4.7.1 Continuous dependence on parameters ............ 131
5 The Discrete Approach and Boundary Behavior ................ 133
5.1 The Euler simulation scheme and its convergence ....... 133
5.2 The pdf of Euler's scheme in and the FPE ............ 137
5.2.1 Unidirectional and net probability flux
density ........................................ 145
5.3 Boundary behavior of diffusions ....................... 150
5.4 Absorbing boundaries .................................. 151
5.4.1 Unidirectional flux and the survival
probability .................................... 155
5.5 Reflecting and partially reflecting boundaries ........ 157
5.5.1 Total and partial reflection in one
dimension ...................................... 158
5.5.2 Partially reflected diffusion in higher
dimensions ..................................... 165
5.5.3 Discontinuous coefficients ..................... 168
5.5.4 Diffusion on a sphere .......................... 168
5.6 The Wiener measure induced by SDEs .................... 169
5.7 Annotations ........................................... 173
6 The First Passage Time of Diffusions ....................... 176
6.1 The FPT and escape from a domain ...................... 176
6.2 The PDF of the FPT .................................... 180
6.3 The exit density and probability flux density ......... 184
6.4 The exit problem in one dimension ..................... 185
6.4.1 The exit time .................................. 191
6.4.2 Application of the Laplace method .............. 194
6.5 Conditioning ......................................... 197
6.5.1 Conditioning on trajectories that reach A
before В ...................................... 198
6.6 Killing measure and the survival probability .......... 202
7 Markov Processes and their Diffusion Approximations ........ 207
7.1 Markov processes ...................................... 207
7.1.1 The general form of the master equation ........ 211
7.1.2 Jump-diffusion processes ....................... 218
7.2 A semi-Markovian example: Renewal processes ........... 222
7.3 Diffusion approximations of Markovian jump processes
Y ..................................................... 230
7.3.1 A refresher on solvability of linear
equations ...................................... 230
7.3.2 Dynamics with large and fast jumps ............. 231
7.3.3 Small jumps and the Kramers-Moyal expansion .... 236
7.3.4 An application to Brownian motion in a field
of force ....................................... 241
7.3.5 Dynamics driven by wideband noise .............. 244
7.3.6 Boundary behavior of diffusion
approximations ................................. 247
7.4 Diffusion approximation of the MFPT .................. 249
8 Diffusion Approximations to Langevin's Equation ............ 257
8.1 The overdamped Langevin equation ...................... 257
8.1.1 The overdamped limit of the GLE ................ 259
8.2 Smoluchowski expansion in the entire space ............ 265
8.3 Boundary conditions in the Smoluchowski limit ......... 268
8.3.1 Appendix ....................................... 275
8.4 Low-friction asymptotics of the FPE ................... 276
8.5 The noisy underdamped forced pendulum ................. 285
8.5.1 The noiseless underdamped forced pendulum ...... 286
8.5.2 Local fluctuations about a nonequilibrium
steady state ................................... 290
8.5.3 The FPE and the MFPT far from equilibrium ...... 295
8.5.4 Application to the shunted Josephson
junction ....................................... 299
8.6 Annotations ........................................... 301
9 Large Deviations of Markovian Jump Processes ............... 302
9.1 The WKB structure of the stationary pdf ............... 302
9.2 The mean time to a large deviation .................... 308
9.3 Asymptotic theory of large deviations ................. 322
9.3.1 More general sums .............................. 328
9.3.2 A central limit theorem for dependent
variables ...................................... 333
9.4 Annotations ........................................... 337
10 Noise-Induced Escape From an Attractor ..................... 339
10.1 Asymptotic analysis of the exit problem ............... 339
10.1.1 The exit problem for small diffusion with the
flow ........................................... 343
10.1.2 Small diffusion against the flow ............... 348
10.1.3 The MFPT of small diffusion against the flow ... 352
10.1.4 Escape over a sharp barrier .................... 353
10.1.5 The MFPT to a smooth boundary and the escape
rate ........................................... 356
10.1.6 The MFPT eigenvalues of the Fokker-Planck
operator ....................................... 359
10.2 The exit problem in higher dimensions ................ 359
10.2.1 The WKB structure of the pdf ................... 361
10.2.2 The eikonal equation ........................... 362
10.2.3 The transport equation ......................... 364
10.2.4 The characteristic equations ................... 365
10.2.5 Boundary layers at noncharacteristic
boundaries ..................................... 366
10.2.6 Boundary layers at characteristic boundaries
in the plane ................................... 369
10.2.7 Exit through noncharacteristic boundaries ...... 371
10.2.8 Exit through characteristic boundaries in
the plane ...................................... 376
10.2.9 Kramers' exit problem .......................... 378
10.3 Activated escape in Langevin's equation ............... 382
10.3.1 The separatrix in phase space .................. 382
10.3.2 Kramers'exit problem at high and low
friction ....................................... 384
10.3.3 The MFPT to the separatrix Г ................... 386
10.3.4 Uniform approximation to Kramers'rate .......... 387
10.3.5 The exit distribution on the separatrix ........ 389
10.4 Annotations ........................................... 397
11 Stochastic Stability ....................................... 399
11.1 Stochastic stability of nonlinear oscillators ......... 403
11.1.1 Underdamped pendulum with parametric noise ..... 404
11.1.2 The steady-state distribution of the noisy
oscillator ..................................... 407
11.1.3 First passage times and stability .............. 410
11.2 Stabilization with oscillations and noise ............. 417
11.2.1 Stabilization by high-frequency noise .......... 417
11.2.2 The generating equation ........................ 418
11.2.3 The correlation-free equation .................. 419
11.2.4 The stability of (11.72) ....................... 421
11.3 Stability of columns with noisy loads ................. 425
11.3.1 A thin column with a noisy load ................ 426
11.3.2 The double pendulum ............................ 429
11.3.3 The damped vertically loaded double pendulum ... 434
11.3.4 A tangentially loaded double pendulum
(follower load) ................................ 436
11.3.5 The N-fold pendulum and the continuous
column ......................................... 438
Bibliography .................................................. 442
Index ......................................................... 459
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