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| 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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