Stability of random attractors under perturbation and approximation
type: Item
instance of: scholarly article
title: Stability of random attractors under perturbation and approximation
zbMATH Open document ID: 1020.37033
DOI: 10.1016/S0022-0396(02)00038-4
author: James C. Robinson
published in: Journal of Differential Equations
publication date: 2003-04-02T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 37H99
Mathematics Subject Classification ID: 37L55
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 37G35
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 37M99
zbMATH DE Number: 1891259
zbMATH Keywords: random attractor
zbMATH Keywords: upper semi-continuous dependence on parameters
zbMATH Keywords: stochastic reaction-diffusion equation
zbMATH Keywords: numerical approximation
zbMATH Keywords: random dynamical systems
reviewed by: Hans Crauel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891259