Limiting dynamics for stochastic wave equations

In this paper, relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered. By introducing almost surely D–α-contracting property for random dynamical systems, we obtain a global random attractor of the stochastic wave equation endowed with Dirichlet boundary condition for any 0<ν⩽1. The upper semicontinuity of this global random attractor and the global attractor of the heat equation zt−Δz+f(z)=0 with Dirichlet boundary condition as ν goes to zero is investigated. Furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation.