| The Physical Brownian Motion: Diffusion And Noise | p. 1 |
| Einstein's theory of diffusion | p. 1 |
| The velocity process and Langevin's approach | p. 5 |
| The displacement process | p. 10 |
| Classical theory of noise | p. 13 |
| An application: Johnson noise | p. 16 |
| Linear systems | p. 21 |
| The Probability Space of Brownian Motion | p. 25 |
| Introduction | p. 25 |
| The space of Brownian trajectories | p. 27 |
| The Wiener measure of Brownian trajectories | p. 37 |
| The MBM in Rd 44 | |
| Constructions of the MBM | p. 46 |
| The Paley-Wiener construction of the Brownian motion | p. 46 |
| P. Lévy's method and refinements | p. 49 |
| Analytical and statistical properties of Brownian paths | p. 52 |
| The Markov property of the MBM | p. 55 |
| Reflecting and absorbing walls | p. 56 |
| MBM and martingales | p. 60 |
| Itô Integration and Calculus | p. 63 |
| Integration of white noise | p. 63 |
| The Itô, Stratonovich, and other integrals | p. 66 |
| The Itô integral | p. 66 |
| The Stratonovich integral | p. 68 |
| The backward integral | p. 73 |
| The construction of the Itô integral | p. 74 |
| The Itô calculus | p. 81 |
| Stochastic Differential Equations | p. 92 |
| Itô and Stratonovich SDEs | p. 93 |
| Transformations of Itô equations | p. 97 |
| Solutions of SDEs are Markovian | p. 101 |
| Stochastic and partial differential equations | p. 104 |
| The Andronov-Vitt-Pontryagin equation | p. 109 |
| The exit distribution | p. 111 |
| The PDF of the FPT | p. 114 |
| The Fokker-Planck equation | p. 119 |
| The backward Kolmogorov equation | p. 124 |
| Appendix: Proof of Theorem 4.1.1 | p. 125 |
| Continuous dependence on parameters | p. 131 |
| The Discrete Approach and Boundary Behavior | p. 133 |
| The Euler simulation scheme and its convergence | p. 133 |
| The pdf of Euler's scheme in R and the FPE | p. 137 |
| Unidirectional and net probability flux density | p. 145 |
| Boundary behavior of diffusions | p. 150 |
| Absorbing boundaries | p. 151 |
| Unidirectional flux and the survival probability | p. 155 |
| Reflecting and partially reflecting boundaries | p. 157 |
| Total and partial reflection in one dimension | p. 158 |
| Partially reflected diffusion in higher dimensions | p. 165 |
| Discontinuous coefficients | p. 168 |
| Diffusion on a sphere | p. 168 |
| The Wiener measure induced by SDEs | p. 169 |
| Annotations | p. 173 |
| The First Passage Time of Diffusions | p. 176 |
| The FPT and escape from a domain | p. 176 |
| The PDF of the FPT | p. 180 |
| The exit density and probability flux density | p. 184 |
| The exit problem in one dimension | p. 185 |
| The exit time | p. 191 |
| Application of the Laplace method | p. 194 |
| Conditioning | p. 197 |
| Conditioning on trajectories that reach A before B | p. 198 |
| Killing measure and the survival probability | p. 202 |
| Markov Processes and their Diffusion Approximations | p. 207 |
| Markov processes | p. 207 |
| The general form of the master equation | p. 211 |
| Jump-diffusion processes | p. 218 |
| A semi-Markovian example: Renewal processes | p. 222 |
| Diffusion approximations of Markovian jump processes | p. 230 |
| A refresher on solvability of linear equations | p. 230 |
| Dynamics with large and fast jumps | p. 231 |
| Small jumps and the Kramers-Moyal expansion | p. 236 |
| An application to Brownian motion in a field of force | p. 241 |
| Dynamics driven by wideband noise | p. 244 |
| Boundary behavior of diffusion approximations | p. 247 |
| Diffusion approximation of the MFPT | p. 249 |
| Diffusion Approximations to Langevin's Equation | p. 257 |
| The overdamped Langevin equation | p. 257 |
| The overdamped limit of the GLE | p. 259 |
| Smohichowski expansion in the entire space | p. 265 |
| Boundary conditions in the Smoluchowski limit | p. 268 |
| Appendix | p. 275 |
| Low-friction asymptotics of the FPE | p. 276 |
| The noisy underdamped forced pendulum | p. 285 |
| The noiseless underdamped forced pendulum | p. 286 |
| Local fluctuations about a nonequilibrium steady state | p. 290 |
| The FPE and the MFPT far from equilibrium | p. 295 |
| Application to the shunted Josephson junction | p. 299 |
| Annotations | p. 301 |
| Large Deviations of Markovian Jump Processes | p. 302 |
| The WKB structure of the stationary pdf | p. 302 |
| The mean time to a large deviation | p. 308 |
| Asymptotic theory of large deviations | p. 322 |
| More general sums | p. 328 |
| A central limit theorem for dependent variables | p. 333 |
| Annotations | p. 337 |
| Noise-Induced Escape From an Attractor | p. 339 |
| Asymptotic analysis of the exit problem | p. 339 |
| The exit problem for small diffusion with the flow | p. 343 |
| Small diffusion against the flow | p. 348 |
| The MFPT of small diffusion against the flow | p. 352 |
| Escape over a sharp barrier | p. 353 |
| The MFPT to a smooth boundary and the escape rate | p. 356 |
| The MFPT eigenvalues of the Fokker-Planck operator | p. 359 |
| The exit problem in higher dimensions | p. 359 |
| The WKB structure of the pdf | p. 361 |
| The eikonal equation | p. 362 |
| The transport equation | p. 364 |
| The characteristic equations | p. 365 |
| Boundary layers at noncharacteristic boundaries | p. 366 |
| Boundary layers at characteristic boundaries in the plane | p. 369 |
| Exit through noncharacteristic boundaries | p. 371 |
| Exit through characteristic boundaries in the plane | p. 376 |
| Kramers'exit problem | p. 378 |
| Activated escape in Langevin's equation | p. 382 |
| The separatrix in phase space | p. 382 |
| Kramers'exit problem at high and low friction | p. 384 |
| The MFPT to the separatrix | p. 386 |
| Uniform approximation to Kramers'rate | p. 387 |
| The exit distribution on the separatrix | p. 389 |
| Annotations | p. 397 |
| Stochastic Stability | p. 399 |
| Stochastic stability of nonlinear oscillators | p. 403 |
| Underdamped pendulum with parametritnoise | p. 404 |
| The steady-state distribution of the noisy oscillator | p. 407 |
| First passage times and stability | p. 410 |
| Stabilization with oscillations and noise | p. 417 |
| Stabilization by high-frequency noise | p. 417 |
| The generating equation | p. 418 |
| The correlation-free equation | p. 419 |
| The stability of (11.72) | p. 421 |
| Stability of columns with noisy loads | p. 425 |
| A thin column with a noisy load | p. 426 |
| The double pendulum | p. 429 |
| The damped vertically loaded double pendulum | p. 434 |
| A tangentially loaded double pendulum (follower load) | p. 436 |
| The N-fold pendulum and the continuous column | p. 438 |
| Bibliography | p. 442 |
| Index | p. 459 |
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