*(English)*Zbl 1020.37033

The paper is concerned with the dependence of the (global) random attractors of a family of random dynamical systems $\left({\varphi}_{\epsilon}(t,\omega )\right)$ on a Polish space $(X,d)$ over a common probability space on a parameter $\epsilon $ (while $\epsilon $ is implicitly assumed to be from $[0,\infty )\subset \mathbb{R}$, the arguments go through more generally). Suppose that one has almost sure pointwise convergence of ${\varphi}_{\epsilon}$ to ${\varphi}_{0}$ in the sense that $P$-almost surely, for every $x\in X$, $d\left({\varphi}_{\epsilon}(t,\omega )y,{\varphi}_{0}(t,\omega )x\right)$ converges to 0 for $\epsilon \to 0$ and $d(x,y)\to 0$. Assuming existence of a random attractor $\omega \mapsto {A}_{\epsilon}\left(\omega \right)$ for ${\varphi}_{\epsilon}$, $\epsilon \in [0,{\epsilon}_{0})$, $P$-almost sure upper semicontinuity of ${A}_{\epsilon}$ in $\epsilon =0$ is shown be equivalent to the existence of a family of attracting sets ${K}_{\epsilon}$ which is upper semicontinuous in $\epsilon =0$ $P$-a. s. Here upper semicontinuity of ${A}_{\epsilon}$ in $\epsilon =0$ means ${lim}_{\epsilon \to 0}\text{dist}({A}_{\epsilon},{A}_{0})=0$, where $\text{dist}$ 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 $\varphi $ by a numerical scheme ${\varphi}_{n}$, $n\in \mathbb{N}$, it is shown that the corresponding assertions proved before to hold almost surely for the attractors of ${\varphi}_{n}$ converging to $\varphi $ 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 $du=({\Delta}u+\beta u-{u}^{3})dt+\sigma u\circ dW\left(t\right)$ with multiplicative noise in dependence of the parameter $\beta $, and to the approximations of the attractor given by a backward Euler approximation of a ${\mathbb{R}}^{d}$-valued stochastic differential equation of the form $dx=f\left(x\right)dt+\epsilon dW\left(t\right)$ under suitable conditions on dissipativity, boundedness and Lipschitz properties of $f$.

##### MSC:

37H99 | Random dynamical systems |

37L55 | Infinite-dimensional random dynamical systems; stochastic equations |

60H10 | Stochastic ordinary differential equations |

37G35 | Attractors and their bifurcations |

60H15 | Stochastic partial differential equations |

37M99 | Approximation methods and numerical treatment of dynamical systems |