The paper is concerned with the dependence of the (global) random attractors of a family of random dynamical systems on a Polish space over a common probability space on a parameter (while is implicitly assumed to be from , the arguments go through more generally). Suppose that one has almost sure pointwise convergence of to in the sense that -almost surely, for every , converges to 0 for and . Assuming existence of a random attractor for , , -almost sure upper semicontinuity of in is shown be equivalent to the existence of a family of attracting sets which is upper semicontinuous in -a. s. Here upper semicontinuity of in means , where denotes the Hausdorff semi-distance.
Next convergence in mean instead of almost sure convergence is discussed. In the context of a numerical approximation of a random dynamical system by a numerical scheme , , it is shown that the corresponding assertions proved before to hold almost surely for the attractors of converging to also hold in mean, provided some additional integrability conditions are satisfied.
The results are then applied to the random attractor of the stochastic reaction-diffusion equation with multiplicative noise in dependence of the parameter , and to the approximations of the attractor given by a backward Euler approximation of a -valued stochastic differential equation of the form under suitable conditions on dissipativity, boundedness and Lipschitz properties of .