The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

✍ Scribed by Arnaud Debussche, Michael Högele, Peter Imkeller (auth.)

Publisher: Springer International Publishing
Year: 2013
Tongue: English
Leaves: 175
Series: Lecture Notes in Mathematics 2085
Edition: 1
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

✦ Table of Contents

Front Matter....Pages i-xiii
Introduction....Pages 1-10
The Fine Dynamics of the Chafee–Infante Equation....Pages 11-43
The Stochastic Chafee–Infante Equation....Pages 45-68
The Small Deviation of the Small Noise Solution....Pages 69-85
Asymptotic Exit Times....Pages 87-120
Asymptotic Transition Times....Pages 121-130
Localization and Metastability....Pages 131-149
Back Matter....Pages 151-165

✦ Subjects

Probability Theory and Stochastic Processes; Dynamical Systems and Ergodic Theory; Partial Differential Equations

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