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Mathematics - Probability Theory and Stochastic Processes | The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

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

Series: Lecture Notes in Mathematics, Vol. 2085

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

Related subjects » Dynamical Systems & Differential Equations - Probability Theory and Stochastic Processes

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