The dynamics of nonlinear reaction-diffusion equations with small Lévy noise

Arnaud Debussche (1, 2) , Michael Högele (3) , Peter Imkeller (4)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 University of Potsdam = Universität Potsdam
4 Institut für Mathematik [Berlin]

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.

Domaines: Probabilités [math.PR]

Arnaud Debussche, Michael Högele, Peter Imkeller. The dynamics of nonlinear reaction-diffusion equations with small Lévy noise. Springer, xiii-163 p., 2013, Lecture notes in mathematics n° 2085, 978-3-319-00827-1. ⟨10.1007/978-3-319-00828-8⟩. ⟨hal-00947868⟩