The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise
Nonfiction, Science & Nature, Mathematics, Mathematical Analysis, Statistics
| Author: | Arnaud Debussche, Michael Högele, Peter Imkeller | ISBN: | 9783319008288 |
| Publisher: | Springer International Publishing | Publication: | October 1, 2013 |
| Imprint: | Springer | Language: | English |
| Author: | Arnaud Debussche, Michael Högele, Peter Imkeller |
| ISBN: | 9783319008288 |
| Publisher: | Springer International Publishing |
| Publication: | October 1, 2013 |
| Imprint: | Springer |
| Language: | English |
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.
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.
