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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 $\bigl(\varphi_\varepsilon(t,\omega)\bigr)$ on a Polish space $(X,d)$ over a common probability space on a parameter $\varepsilon$ (while $\varepsilon$ is implicitly assumed to be from $[0,\infty)\subset\bbfR$, the arguments go through more generally). Suppose that one has almost sure pointwise convergence of $\varphi_\varepsilon$ to $\varphi_0$ in the sense that $P$-almost surely, for every $x\in X$, $d\bigl(\varphi_\varepsilon(t,\omega)y,\varphi_0(t,\omega)x\bigr)$ converges to $0$ for $\varepsilon\to 0$ and $d(x,y)\to 0$. Assuming existence of a random attractor $\omega\mapsto A_\varepsilon(\omega)$ for $\varphi_\varepsilon$, $\varepsilon\in[0,\varepsilon_0)$, $P$-almost sure upper semicontinuity of $A_\varepsilon$ in $\varepsilon=0$ is shown be equivalent to the existence of a family of attracting sets $K_\varepsilon$ which is upper semicontinuous in $\varepsilon=0$ $P$-a. s. Here upper semicontinuity of $A_\varepsilon$ in $\varepsilon=0$ means $\lim_{\varepsilon\to 0}\text{dist} (A_\varepsilon,A_0)=0$, 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 $\varphi_n$, $n\in\bbfN$, 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 $du=(\Delta u+\beta u-u^3) dt+\sigma u\circ dW(t)$ with multiplicative noise in dependence of the parameter $\beta$, and to the approximations of the attractor given by a backward Euler approximation of a $\bbfR^d$-valued stochastic differential equation of the form $dx=f(x) dt+\varepsilon dW(t)$ 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
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