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

  • The comprehensive presentation serves as an excellent basis for a Master's course on stochastic partial differential equations(SPDEs) with Lévy noise

  • The showcase character of this study provides particular insight into the methods developed and stimulates future research

  • An additional chapter connects the mathematical results to its climatological motivation

  • Includes supplementary material: sn.pub/extras

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

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xiii
  2. Introduction

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 1-10
  3. The Fine Dynamics of the Chafee–Infante Equation

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 11-43
  4. The Stochastic Chafee–Infante Equation

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 45-68
  5. The Small Deviation of the Small Noise Solution

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 69-85
  6. Asymptotic Exit Times

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 87-120
  7. Asymptotic Transition Times

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 121-130
  8. Localization and Metastability

    • Arnaud Debussche, Michael Högele, Peter Imkeller
    Pages 131-149
  9. Back Matter

    Pages 151-165

About this book

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
  • partial differential equations

Authors and Affiliations

  • Ecole Normale Supérieure Cachan Antenne de Bretagne, Bruz, Rennes, France

    Arnaud Debussche

  • Institut für Mathematik LS Wahrscheinlichkeitstheorie, Universität Potsdam, Potsdam, Germany

    Michael Högele

  • Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Peter Imkeller

Bibliographic Information

  • Book Title: The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

  • Authors: Arnaud Debussche, Michael Högele, Peter Imkeller

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-00828-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2013

  • Softcover ISBN: 978-3-319-00827-1Published: 14 October 2013

  • eBook ISBN: 978-3-319-00828-8Published: 01 October 2013

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XIV, 165

  • Number of Illustrations: 1 b/w illustrations, 8 illustrations in colour

  • Topics: Probability Theory, Dynamical Systems, Differential Equations

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions