The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise
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 ; . - Lecture Notes in Mathematics, 2085 .
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
10.1007/978-3-319-00828-8 doi
Distribution (Probability theory.
Differentiable dynamical systems.
Differential equations, partial.
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 ; . - Lecture Notes in Mathematics, 2085 .
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
10.1007/978-3-319-00828-8 doi
Distribution (Probability theory.
Differentiable dynamical systems.
Differential equations, partial.
Probability Theory and Stochastic Processes.
Dynamical Systems and Ergodic Theory.
Partial Differential Equations.
QA273.A1-274.9 QA274-274.9
519.2