Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in ApplicationsAsymptotic methods are of great importance for practical applications, especially in dealing with boundary value problems for small stochastic perturbations. This book deals with nonlinear dynamical systems perturbed by noise. It addresses problems in which noise leads to qualitative changes, escape from the attraction domain, or extinction in population dynamics. The most likely exit point and expected escape time are determined with singular perturbation methods for the corresponding Fokker-Planck equation. The authors indicate how their techniques relate to the Itô calculus applied to the Langevin equation. The book will be useful to researchers and graduate students. |
Contents
the Langevin Equation | 3 |
12 The Ito Calculus | 7 |
13 Small Noise Expansion of the Langevin Equation | 10 |
14 Simulation of the Stochastic Process | 12 |
15 Exercises | 13 |
First Exit from a Domain | 18 |
22 The Exit Probability and the Expected Exit Time | 23 |
23 Exercises | 25 |
55 Exercises | 95 |
6 Dispersive Groundwater Flow and Pollution | 99 |
61 The Boundary Layer for a Symmetric Flow Field | 101 |
62 The Boundary Layer for an Arbitrary Flow Field | 107 |
of Interacting Biological Populations | 118 |
72 The SIRModel in Stochastic Epidemiology | 130 |
73 Extinction of a Population Within a System of Interacting Populations | 141 |
8 Stochastic Oscillation | 149 |
One Dimension | 27 |
31 Stationary and QuasiStationary Distributions | 28 |
32 Exit Time and Exit Probability | 32 |
33 Exercises | 38 |
4 Singular Perturbation Analysis of the Differential Equations for the Exit Probability and Exit Time in One Dimension | 43 |
42 The Expected Exit Time | 50 |
43 Vanishing Diffusion and Drift at a Boundary | 52 |
44 The Problem of Unlikely Exit Using the WKBMethod | 57 |
45 Exercises | 70 |
in Several Dimensions the Asymptotic Exit Problem | 73 |
51 Exit by Diffusion Across the Drift | 74 |
52 Exit by Diffusion Along the Drift | 78 |
53 Exit by Diffusion Against the Drift | 80 |
54 Exit from the Domain of Attraction | 91 |
81 Equivalent Statistical Linearization | 150 |
82 Almost Linear Oscillation and Stochastic Averaging | 152 |
83 Stochastic Relaxation Oscillation | 156 |
9 Confidence Domain Return Time and Control | 168 |
92 Return Time of a Stochastic System and Its Application in Ecology | 171 |
93 Applications in Control Theory | 180 |
10 A Markov Chain Approximation of the Stochastic Dynamical System | 184 |
102 Extinction and Recolonization in Population Biology | 192 |
| 203 | |
Answers to Exercises | 211 |
| 215 | |
| 219 | |
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Common terms and phrases
a₁ advective analysis asymptotic approximation b₁ behaviour boundary conditions boundary layer boundary value problem boundary x2 computed constant coordinate d₁ derive deterministic system diffusion divergence theorem domain of attraction dr² drift dW(t dX₁ dynamical system eigenvalues exit boundary exit probability Exit Problem expected arrival expected exit expected extinction expected value ɛdW(t flow Fokker-Planck equation Gaussian given Grasman Herwaarden integral interval 0,1 Itô calculus K₂ Langevin Equation linear matrix N₁ N₂ neighbourhood nonlinear obtain order approximation oscillator parameter particle population prey-predator system probability density function probability of exit random reflecting boundary satisfying Sect separating streamline simulation singular perturbation spectral model stable equilibrium stagnation point starting points stationary distribution stochastic differential equations stochastic logistic stochastic process stochastic process dX Substitution T₁(x Tbound tion trajectory variable Wiener process x₁ x₂ yields ε² ΘΩ ди др дх эт