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Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications : Springer Series in Synergetics - Johan Grasman

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

Springer Series in Synergetics

Hardcover

Published: 8th March 1999
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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 It“ calculus applied to the Langevin equation. The book will be useful to researchers and graduate students.

The Fokker-Planck Equation
Dynamical Systems Perturbed by Noise: the Langevin Equationp. 3
Stochastic Processes and the Effect of Small Noisep. 3
The Itô Calculusp. 7
Small Noise Expansion of the Langevin Equationp. 10
Simulation of the Stochastic Processp. 12
Exercisesp. 13
The Fokker-Planck Equation: First Exit from a Domainp. 18
The Forward and the Backward Equationp. 18
The Exit Probability and the Expected Exit Timep. 23
Exercisesp. 25
The Fokker-Planck Equation: One Dimensionp. 27
Stationary and Quasi-Stationary Distributionsp. 28
Exit Time and Exit Probabilityp. 32
Exercisesp. 38
Asymptotic Solution of the Exit Problem
Singular Perturbation Analysis of the Differential Equations for the Exit Probability and Exit Time in One Dimensionp. 43
The Exit Probabilityp. 43
The Expected Exit Timep. 50
Vanishing Diffusion and Drift at a Boundaryp. 52
The Problem of Unlikely Exit Using the WKB-Methodp. 57
Exercisesp. 70
The Fokker-Planck Equation in Several Dimensions: the Asymptotic Exit Problemp. 73
Exit by Diffusion Across the Driftp. 74
Exit by Diffusion Along the Driftp. 78
Exit hy Diffusion Against the Driftp. 80
Exit from the Domain of Attractionp. 91
Exercisesp. 95
Applications
Dispersive Groundwater Flow and Pollutionp. 99
The Boundary Layer for a Symmetric Flow Fieldp. 101
The Boundary Layer for an Arbitrary Flow Fieldp. 107
Extinction in Systems of Interacting Biological Populationsp. 118
A Prey-Predator Systemp. 118
The SIR-Model in Stochastic Epidemiologyp. 130
Extinction of a Population Within a System of Interacting Populationsp. 141
Stochastic Oscillationp. 149
Equivalent Statistical Linearizationp. 150
Almost Linear Oscillation and Stochastic Averagingp. 152
Stochastic Relaxation Oscillationp. 156
Confidence Domain, Return Time and Controlp. 168
Confidence Domainp. 168
Retum Time of a Stochastic System and Its Application in Ecologyp. 171
Applications in Control Theoryp. 180
A Markov Chain Approximation of the Stochastic Dynamical Systemp. 184
Preferent States in a Low Order Spectral Model of the Atmospheric Circulationp. 184
Extinction and Recolonization in Population Biologyp. 192
Literaturep. 203
Answers to Exercisesp. 211
Author Indexp. 215
Subject Indexp. 219
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540644354
ISBN-10: 3540644350
Series: Springer Series in Synergetics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 220
Published: 8th March 1999
Publisher: SPRINGER VERLAG GMBH
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.42
Weight (kg): 0.51