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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 ϕ ε (t,ω) on a Polish space (X,d) over a common probability space on a parameter ε (while ε is implicitly assumed to be from [0,), the arguments go through more generally). Suppose that one has almost sure pointwise convergence of ϕ ε to ϕ 0 in the sense that P-almost surely, for every xX, dϕ ε (t,ω) y , ϕ 0 (t,ω) x converges to 0 for ε0 and d(x,y)0. Assuming existence of a random attractor ωA ε (ω) for ϕ ε , ε[0,ε 0 ), P-almost sure upper semicontinuity of A ε in ε=0 is shown be equivalent to the existence of a family of attracting sets K ε which is upper semicontinuous in ε=0 P-a. s. Here upper semicontinuity of A ε in ε=0 means lim ε0 dist(A ε ,A 0 )=0, where 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 ϕ by a numerical scheme ϕ n , n, it is shown that the corresponding assertions proved before to hold almost surely for the attractors of ϕ n converging to ϕ 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=(Δu+βu-u 3 )dt+σudW(t) with multiplicative noise in dependence of the parameter β, and to the approximations of the attractor given by a backward Euler approximation of a d -valued stochastic differential equation of the form dx=f(x)dt+εdW(t) under suitable conditions on dissipativity, boundedness and Lipschitz properties of f.


MSC:
37H99Random dynamical systems
37L55Infinite-dimensional random dynamical systems; stochastic equations
60H10Stochastic ordinary differential equations
37G35Attractors and their bifurcations
60H15Stochastic partial differential equations
37M99Approximation methods and numerical treatment of dynamical systems