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Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. (English) Zbl 1188.37076

The authors introduce the notion of quasi-continuity for a dynamical system $\varphi$ on a Banach space $B$ to mean that for any sequence $\left({t}_{n},{x}_{n}\right)$ converging to $\left(t,x\right)$ in $\left[0,\infty \right)×B$, for which ${\left(\varphi \left({t}_{n}\right){x}_{n}\right)}_{n\in ℕ}$ is bounded, $\varphi \left({t}_{n}\right){x}_{n}$ converges weakly to $\varphi \left(t\right)x$. This property is weaker than norm-to-weak continuity. The authors invoke this notion for random dynamical systems on a separable $B$ in order to derive a sufficient condition for the existence of a random attractor, using asymptotic compactness properties formulated in terms of the Kuratowski measure of non-compactness.

The result is applied to stochastic reaction-diffusion equations on $D\subset {ℝ}^{d}$ bounded, with a finite-dimensional additive Wiener process taking values in ${L}^{\infty }\left(D\right)$. The nonlinearity of the drift being polynomially bounded, one has to consider solutions in ${L}^{p}\left(D\right)$ for suitable $p>2$. The induced random dynamical system is quasi-continuous, but it fails to be norm-to-weak continuous. It is shown to have a finite-dimensional random attractor $A$ for bounded subsets of ${L}^{2}\left(D\right)$, where attraction holds with respect to the ${L}^{p}\left(D\right)$ norm for every $p\ge 2$. Finally, a comparison estimate for the fractal dimensions of $A$ with respect to ${L}^{p}\left(D\right)$ norms, $p\ge 2$, is given.

##### MSC:
 37L55 Infinite-dimensional random dynamical systems; stochastic equations 35B41 Attractors (PDE) 35R60 PDEs with randomness, stochastic PDE 37L30 Attractors and their dimensions, Lyapunov exponents 60H15 Stochastic partial differential equations