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Contents
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 Rd
. . . . . . . . . . . . . . . . . . . . . . . 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 Itô 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
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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 . . . . . . . . . . . . . . . . . . . . . 111
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 R 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 B . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.3.1 A refresher on solvability of linear equations . . . . . . . . 230
7.3.2 Dynamics with large and fast jumps . . . . . . . . . . . . . 231
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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
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