On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations mu u(tt)(t,x)=Delta u(t,x)-u(t)(t,x)+b(x,u(t,x))+QW(t),u(0)=u(0), u(t)(0)=v(0), endowed with Dirichlet boundary conditions, converges as mu 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))+QW (t),u(0)=u(0), endowed with Dirichlet boundary conditions. Moreover we consider relations between asymptotics for the heat and for the wave equation. More precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation, for any mu > 0.