Abstract

Co-authors: Arnaud Debussche (ENS Cachan), Jan Gairing (HU Berlin), Claudia Hein (HU Berlin), Michael Hgele (U Potsdam), Ilya Pavlyukevich (U Jena)

Dynamical systems of the reaction-diffusion type with small noise have been instrumental to explain basic features of the dynamics of paleo-climate data. For instance, a spectral analysis of Greenland ice time series performed at the end of the 1990s representing average temperatures during the last ice age suggest an $lpha-$stable noise component with an $lpha im 1.75.$ We model the time series as a dynamical system perturbed by $lpha$-stable noise, and develop an efficient testing method for the best fitting $lpha$. The method is based on the observed $p$-variation of the residuals of the time series, and their asymptotic $rac{lpha}{p}$-stability established in local limit theorems.par mallskip

Generalizing the solution of this model selection problem, we are led to a class of reaction-diffusion equations with additive $lpha$-stable L’evy noise, a stochastic perturbation of the Chafee-Infante equation. We study exit and transition between meta-stable states of their solutions. Due to the heavy-tail nature of an $lpha$-stable noise component, the results differ strongly from the well known case of purely Gaussian perturbations.