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Smoluchowski-Kramers approximation for a general class of SPDEs

Smoluchowski-Kramers approximation for a general class of SPDEs We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations $$\mu u_{tt} (t,x) + u_{t} (t,x) = \Delta u(t,x) + b(x,u(t,x)) + g(x,u(t,x))\dot{w}(t) $$ , u (0) = u 0 , u t (0) = v 0 , endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation $$u_{t} (t,x) = \Delta u(t,x) + b(x,u(t,x)) + g(x,u(t,x))\dot{w}(t) $$ , u (0) = u 0 , endowed with Dirichlet boundary conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Smoluchowski-Kramers approximation for a general class of SPDEs

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References (20)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Birkhäuser Verlag, Basel
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-006-0281-8
Publisher site
See Article on Publisher Site

Abstract

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations $$\mu u_{tt} (t,x) + u_{t} (t,x) = \Delta u(t,x) + b(x,u(t,x)) + g(x,u(t,x))\dot{w}(t) $$ , u (0) = u 0 , u t (0) = v 0 , endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation $$u_{t} (t,x) = \Delta u(t,x) + b(x,u(t,x)) + g(x,u(t,x))\dot{w}(t) $$ , u (0) = u 0 , endowed with Dirichlet boundary conditions.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Dec 1, 2006

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