The dynamics of nonlinear reaction-diffusion equations with small Lévy noise
type: Item
instance of: scholarly article
title: The dynamics of nonlinear reaction-diffusion equations with small Lévy noise
zbMATH Open document ID: 1321.60004
DOI: 10.1007/978-3-319-00828-8
author: A. Debussche
author: P. Imkeller
author: Michael Högele
published in: Lecture Notes in Mathematics
publication date: 2013-06-13T00:00:00Z
review text: These lecture notes investigate the stochastically perturbed Chafee-Infante equation \[ du=\bigl(\frac{\partial^2}{\partial x^2}u+\lambda(u-u^3)\bigr)\,dt +\varepsilon dL(t),\qquad x\in(0,1),\;t>0, \eqno(*) \] with \(u(t,0)=u(t,1)=0\), where~\(L\) is a pure jump space-time Lévy process and~\(\lambda\) is a parameter with \(0<\lambda\neq k^2\pi^2\) for \(k\in\mathbb N\), for small \(\varepsilon\). The unperturbed system \(\varepsilon=0\) has a unique stable fixed point for \(0<\lambda<\pi^2\) and two stable fixed points together with a number of unstable fixed points for \(\pi^2<\lambda\), increasing as \(\lambda\) passes~\(k^2\pi^2\), \(k\geq2\). These fixed points together with their unstable manifolds form an attracting set. The major results state that the exit and transition times between domains of attraction depend polynomially on the noise intensity~\(\varepsilon\), provided that some non-degeneracy conditions on the Lévy process \(L\) are satisfied. Furthermore, the solutions behave in a metastable way insofar as they can be approximated by a finite-state Markov chain on a suitable polynomial time scale. The polynomial dependence on~\(\varepsilon\) in the case of a Lévy perturbation is substantially different from the well-investigated case of Gaussian perturbations, where the dependence on~\(\varepsilon\) is exponential. The presentation is organized as follows: after an introduction with a detailed outline of the results, Chapter 2 provides a discussion of the deterministic Chafee-Infante equation \(\varepsilon=0\). Chapter 3 describes the Lévy process, giving a decomposition into ``large'' and ``small jump'' components, and a brief introduction into the basic existence and uniqueness results for solutions of \((*)\). Chapter 4 provides small deviation results of the solution of the stochastic equation with only ``small jump'' components from the deterministic solution. In Chapter 5, probabilities of exit events from neighbourhoods of the stable fixed poin…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 86A10
zbMATH DE Number: 6175547
zbMATH Keywords: stochastically perturbed Chafee-Infante equation
zbMATH Keywords: additive Lévy noise
zbMATH Keywords: metastability
zbMATH Keywords: exit time
zbMATH Keywords: transition time
zbMATH Keywords: climate dynamics
reviewed by: Hans Crauel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-00828-8
OpenAlex ID: W368946631
description: scientific article; zbMATH DE number 6175547