ISBN: 3-540-64435-0
TITLE: Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications
AUTHOR: Grasman, Johan; Herwaarden, Onno A., van
TOC:
Part I The Fokker-Planck Equation
1. Dynamical Systems Perturbed by Noise:
the Langevin Equation 3
1.1 Stochastic Processes and the Effect of Small Noise 3
1.2 The Itô Calculus 7
1.3 Small Noise Expansion of the Langevin Equation 10
1.4 Simulation of the Stochastic Process 12
1.5 Exercises 13
2. The Fokker-Planck Equation: First Exit from a Domain 18
2.1 The Forward and the Backward Equation 18
2.2 The Exit Probability and the Expected Exit Time 23
2.3 Exercises 25
3. The Fokker-Planck Equation: One Dimension 27
3.1 Stationary and Quasi-Stationary Distributions 28
3.2 Exit Time and Exit Probability 32
3.3 Exercises 38
Part II Asymptotic Solution of the Exit Problem
4. Singular Perturbation Analysis of the Differential Equations
for the Exit Probability and Exit Time in One Dimension 43
4.1 The Exit Probability 43
4.2 The Expected Exit Time 50
4.3 Vanishing Diffusion and Drift at a Boundary 52
4.4 The Problem of Unlikely Exit Using the WKB-Method 57
4.5 Exercises 70
5. The Fokker-Planck Equation in Several Dimensions:
the Asymptotic Exit Problem 73
5.1 Exit by Diffusion Across the Drift 74
5.2 Exit by Diffusion Along the Drift 78
5.3 Exit by Diffusion Against the Drift 80
5.4 Exit from the Domain of Attraction 91
5.5 Exercises 95
Part III Applications
6. Dispersive Groundwater Flow and Pollution 99
6.1 The Boundary Layer for a Symmetric Flow Field 101
6.2 The Boundary Layer for an Arbitrary Flow Field 107
7. Extinction in Systems
of Interacting Biological Populations 118
7.1 A Prey-Predator System 118
7.2 The SIR-Model in Stochastic Epidemiology 130
7.3 Extinction of a Population
Within a System of Interacting Populations 141
8. Stochastic Oscillation 149
8.1 Equivalent Statistical Linearization 150
8.2 Almost Linear Oscillation and Stochastic Averaging 152
8.3 Stochastic Relaxation Oscillation 156
9. Confidence Domain, Return Time and Control 168
9.1 Confidence Domain 168
9.2 Retum Time of a Stochastic System
and Its Application in Ecology 171
9.3 Applications in Control Theory 180
10. A Markov Chain Approximation of the Stochastic
Dynamical System 184
10.1 Preferent States in a Low Order Spectral Model
of the Atmospheric Circulation 184
10.2 Extinction and Recolonization in Population Biology 192
Literature 203
Answers to Exercises 211
Author Index 215
Subject Index 219
END