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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 of Evolution Equations – Springer Journals
Published: Dec 1, 2006
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