Resumen
In this article we study the behavior of dissipative systems with additive fractional noise of any Hurst parameter. Under a one-sided dissipative Lipschitz condition on the drift the continuous stochastic system is shown to have a unique stationary solution, which pathwise attracts all other solutions. The same holds for the discretized stochastic system, if the drift-implicit Euler method is used for the discretization. Moreover, the unique stationary solution of the drift-implicit Euler scheme converges to the unique stationary solution of the original system as the stepsize of the discretization decreases.

Resumen
We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities. [ABSTRACT FROM AUTHOR]