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9783540644354

Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications

by ;
  • ISBN13:

    9783540644354

  • ISBN10:

    3540644350

  • Format: Hardcover
  • Copyright: 1999-01-01
  • Publisher: Springer Verlag

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Summary

Asymptotic 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.

Table of Contents

Part I The Fokker--Planck Equation
Dynamical Systems Perturbed by Noise: the Langevin Equation
3(15)
Stochastic Processes and the Effect of Small Noise
3(4)
The Ito Calculus
7(3)
Small Noise Expansion of the Langevin Equation
10(2)
Simulation of the Stochastic Process
12(1)
Exercises
13(5)
The Fokker--Planck Equation: First Exist from a Domain
18(9)
The Forward and the Backward Equation
18(5)
The Exit Probability and the Expected Exit Time
23(2)
Exercises
25(2)
The Fokker--Planck Equation: One Dimension
27(16)
Stationary and Quasi-Stationary Distributions
28(4)
Exit Time and Exit Probability
32(6)
Exercises
38(5)
Part II Asymptotic Solution of the Exit Problem
Singular Perturbation Analysis of the Differential Equations for the Exit Probability and Exit Time in One Dimension
43(30)
The Exit Probability
43(7)
The Expected Exit Time
50(2)
Vanishing Diffusion and Drift at a Boundary
52(5)
The Problem of Unlikely Exit Using the WKB-Method
57(13)
Exercises
70(3)
The Fokker--Planck Equation in Several Dimensions: the Asymptotic Exit Problem
73(26)
Exit by Diffusion Across the Drift
74(4)
Exit by Diffusion Along the Drift
78(2)
Exit by Diffusion Against the Drift
80(11)
Exit from the Domain of Attraction
91(4)
Exercises
95(4)
Part III Applications
Dispersive Groundwater Flow and Pollution
99(19)
The Boundary Layer for a Symmetric Flow Field
101(6)
The Boundary Layer for an Arbitrary Flow Field
107(11)
Extinction in Systems of Interacting Biological Populations
118(31)
A Prey-Predator System
118(12)
The SIR-Model in Stochastic Epidemiology
130(11)
Extinction of a Population Within a System of Interacting Populations
141(8)
Stochastic Oscillation
149(19)
Equivalent Statistical Linearization
150(2)
Almost Linear Oscillation and Stochastic Averaging
152(4)
Stochastic Relaxation Oscillation
156(12)
Confidence Domain, Return Time and Control
168(16)
Confidence Domain
168(3)
Return Time of a Stochastic System and Its Application in Ecology
171(9)
Applications in Control Theory
180(4)
A Markov Chain Approximation of the Stochastic Dynamical System
184(19)
Preferent States in a Low Order Spectral Model of the Atmospheric Circulation
184(8)
Extinction and Recolonization in Population Biology
192(11)
Literature 203(8)
Answers to Exercises 211(4)
Author Index 215(4)
Subject Index 219

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