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Stability of random attractors under perturbation and approximation. (English) Zbl 1020.37033

The paper is concerned with the dependence of the (global) random attractors of a family of random dynamical systems $\left({\varphi }_{\epsilon }\left(t,\omega \right)\right)$ on a Polish space $\left(X,d\right)$ over a common probability space on a parameter $\epsilon$ (while $\epsilon$ is implicitly assumed to be from $\left[0,\infty \right)\subset ℝ$, 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 $x\in X$, $d\left({\varphi }_{\epsilon }\left(t,\omega \right)y,{\varphi }_{0}\left(t,\omega \right)x\right)$ converges to 0 for $\epsilon \to 0$ and $d\left(x,y\right)\to 0$. Assuming existence of a random attractor $\omega ↦{A}_{\epsilon }\left(\omega \right)$ for ${\varphi }_{\epsilon }$, $\epsilon \in \left[0,{\epsilon }_{0}\right)$, $P$-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 ${lim}_{\epsilon \to 0}\text{dist}\left({A}_{\epsilon },{A}_{0}\right)=0$, 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 ${\varphi }_{n}$, $n\in ℕ$, 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 $du=\left({\Delta }u+\beta u-{u}^{3}\right)dt+\sigma u\circ dW\left(t\right)$ with multiplicative noise in dependence of the parameter $\beta$, and to the approximations of the attractor given by a backward Euler approximation of a ${ℝ}^{d}$-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