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Debussche, Arnaud.
The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise [electronic resource] / by Arnaud Debussche, Michael Högele, Peter Imkeller. - XIV, 165 p. 9 illus., 8 illus. in color. online resource. - Lecture Notes in Mathematics, 2085 0075-8434 ; .
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.
9783319008288
Mathematics.
Differentiable dynamical systems.
Differential equations, partial.
Distribution (Probability theory).
Mathematics.
Probability Theory and Stochastic Processes.
Dynamical Systems and Ergodic Theory.
Partial Differential Equations.
QA273.A1-274.9 QA274-274.9
519.2
The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise [electronic resource] / by Arnaud Debussche, Michael Högele, Peter Imkeller. - XIV, 165 p. 9 illus., 8 illus. in color. online resource. - Lecture Notes in Mathematics, 2085 0075-8434 ; .
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.
9783319008288
Mathematics.
Differentiable dynamical systems.
Differential equations, partial.
Distribution (Probability theory).
Mathematics.
Probability Theory and Stochastic Processes.
Dynamical Systems and Ergodic Theory.
Partial Differential Equations.
QA273.A1-274.9 QA274-274.9
519.2