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

  • Arnaud Debussche
  • Michael Högele
  • Peter Imkeller

Part of the Lecture Notes in Mathematics book series (LNM, volume 2085)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 1-10
  3. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 11-43
  4. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 45-68
  5. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 69-85
  6. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 87-120
  7. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 121-130
  8. Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 131-149
  9. Back Matter
    Pages 151-165

About this book

Introduction

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.

Keywords

Conceptual climate models First exit problem Metastability Non-Gaussian Lévy noise Stochastic nonlinear reaction-diffusion equations

Authors and affiliations

  • Arnaud Debussche
    • 1
  • Michael Högele
    • 2
  • Peter Imkeller
    • 3
  1. 1.Ecole Normale Supérieure Cachan Antenne de BretagneBruz, RennesFrance
  2. 2.Institut für Mathematik LS WahrscheinlichkeitstheorieUniversität PotsdamPotsdamGermany
  3. 3.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-00828-8
  • Copyright Information Springer International Publishing Switzerland 2013
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-00827-1
  • Online ISBN 978-3-319-00828-8
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book