Smoluchowski-Kramers approximation for a general class of SPDEs

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) = u0, ut (0) = v0, 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) = u0, endowed with Dirichlet boundary conditions.