The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise
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Beschreibung
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.
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.
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 Includes supplementary material: sn.pub/extras
Autor*in
Arnaud Debussche
Themen in »The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise«
Conceptual climate models First exit problem Metastability Non-Gaussian Lévy noise Stochastic nonlinear reaction-diffusion equations partial differential equations