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

• Main Authors: Debussche, Arnaud. (Author), Hg̲ele, Michael. (Author), Imkeller, Peter. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2013.
• Series: Lecture Notes in Mathematics, 2085
• Subjects: Mathematics.; Differentiable dynamical systems.; Differential equations, partial.; Distribution (Probability theory).; Probability Theory and Stochastic Processes.; Dynamical Systems and Ergodic Theory.; Partial Differential Equations.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-3-319-00828-8
• Physical Description: XIII, 163 p. 9 illus., 8 illus. in color. online resource.
• ISBN: 9783319008288
• ISSN: 0075-8434

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.