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-alpha-contracting property for random dynamical systems, we obtain a global random attractor of the stochastic wave equation vu(tt)(v) + u(t)(v) - Delta u(v) + f (u(v)) = root v(W) over dot endowed with Dirichlet boundary condition for any 0 < v <= 1. The upper semicontinuity of this global random attractor and the global attractor of the heat equation z(t) - Delta z + f (z) = 0 with Dirichlet boundary condition as v 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. (C) 2007 Elsevier Inc. All rights reserved.