Lévy Noise Induced Transitions Between Meta-Stable States in Stochastic (Partial) Differential Equations

Peter Imkeller

A spectral analysis of the time series representing average temperatures during the last ice age featuring the Dansgaard-Oeschger events reveals an a-stable noise component with an a ~ 1.78. Based on this observation, papers in the physics literature attempted a qualitative interpretation by studying diffusion equations that describe simple dynamical systems perturbed by small Lévy noise. We study exit and transition problems for solutions of stochastic differential equations and stochastic reaction-diffusion equations derived from this proto type. Due to the heavy-tail nature of the a-stable component of the noise, the results differ strongly from the well known case of purely Gaussian perturbations. For SPDE, transitions are governed by the modes with the largest jumps.

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