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Paleo-climatic time series: statistics and dynamics

Tuesday 29th October 2013 - 10:10 to 10:45
INI Seminar Room 1
Co-authors: Arnaud Debussche (ENS Cachan), Jan Gairing (HU Berlin), Claudia Hein (HU Berlin), Michael Högele (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 $\alpha-$stable noise component with an $\alpha\sim 1.75.$ We model the time series as a dynamical system perturbed by $\alpha$-stable noise, and develop an efficient testing method for the best fitting $\alpha$. The method is based on the observed $p$-variation of the residuals of the time series, and their asymptotic $\frac{\alpha}{p}$-stability established in local limit theorems.\par\smallskip

Generalizing the solution of this model selection problem, we are led to a class of reaction-diffusion equations with additive $\alpha$-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 $\alpha$-stable noise component, the results differ strongly from the well known case of purely Gaussian perturbations.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons