Debussche, Arnaud, Högele, Michael, Imkeller, Peter
2013, XIV, 165 p. 9 illus., 8 illus. in color.
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The comprehensive presentation serves as an excellent basis for a Master's course on stochastic partial differential equations(SPDEs) with Lévy noise
The showcase character of this study provides particular insight into the methods developed and stimulates future research
An additional chapter connects the mathematical results to its climatological motivation
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.
Content Level »Research
Keywords »Conceptual climate models - First exit problem - Metastability - Non-Gaussian Lévy noise - Stochastic nonlinear reaction-diffusion equations