Lecture Notes in Mathematics

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

Authors: Debussche, Arnaud, Högele, Michael, Imkeller, Peter

  • 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
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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.

Table of contents (7 chapters)

  • Introduction

    Debussche, Arnaud (et al.)

    Pages 1-10

  • The Fine Dynamics of the Chafee–Infante Equation

    Debussche, Arnaud (et al.)

    Pages 11-43

  • The Stochastic Chafee–Infante Equation

    Debussche, Arnaud (et al.)

    Pages 45-68

  • The Small Deviation of the Small Noise Solution

    Debussche, Arnaud (et al.)

    Pages 69-85

  • Asymptotic Exit Times

    Debussche, Arnaud (et al.)

    Pages 87-120

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-3-319-00828-8
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA
  • ISBN 978-3-319-00827-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2085
Copyright
2013
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-00828-8
DOI
10.1007/978-3-319-00828-8
Softcover ISBN
978-3-319-00827-1
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XIV, 165
Number of Illustrations and Tables
1 b/w illustrations, 8 illustrations in colour
Topics