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The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

by Debussche, Arnaud.
Authors: Högele, Michael.%author. | Imkeller, Peter.%author. | SpringerLink (Online service) Series: Lecture Notes in Mathematics, 0075-8434 ; . 2085 Physical details: XIV, 165 p. 9 illus., 8 illus. in color. online resource. ISBN: 3319008285 Subject(s): Mathematics. | Differentiable dynamical systems. | Differential equations, partial. | Distribution (Probability theory). | Mathematics. | Probability Theory and Stochastic Processes. | Dynamical Systems and Ergodic Theory. | Partial Differential Equations.
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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.

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