RESUMEN

Under one-sided dissipative Lipschitz condition its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Galerkin system as the stepsize tends to zero and the stationary solution of the Galerkin system converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of  appropriate Ornstein-Uhlenbeck stationary solutions.