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The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise
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ISBN-13:
9783319008271
Veröffentl:
2013
Einband:
Paperback
Erscheinungsdatum:
14.10.2013
Seiten:
180
Autor:
Arnaud Debussche
Gewicht:
283 g
Format:
235x155x11 mm
Serie:
2085, Lecture Notes in Mathematics
Sprache:
Englisch
Beschreibung:
The comprehensive presentation serves as an excellent basis for a Master's course on stochastic partial differential equations(SPDEs) with Lévy noise
Introduction.- The fine dynamics of the Chafee- Infante equation.- The stochastic Chafee- Infante equation.- The small deviation of the small noise solution.- Asymptotic exit times.- Asymptotic transition times.- Localization and metastability.- The source of stochastic models in conceptual climate dynamics.
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.