Stability of random attractors under perturbation and approximation
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 \([0,\infty)\subset\mathbb{R}\), 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~\(\varphi_n\), \(n\in\mathbb{N}\), 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 \(du=(\Delta u+\beta u-u^3) dt+\sigma u\circ dW(t)\) with multiplicative noise in dependence of the parameter~\(\beta\), and to the approximations of the attractor given by a backward Euler approximation of a \(\mathbb{R}^d\)-valued stochastic differential equation of the form \(dx=f(x) dt+\varepsilon dW(t)\) under suitable conditions on dissipativity, boundedness and Lipschitz properties of~\(f\).
upper semi-continuous dependence on parameters
scientific article; zbMATH DE number 1891259