Random attractors for dissipative systems with rough noises
Hoang Duc Luu
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Submission date: 06. Oct. 2020 (revised version: September 2021)
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We provide an analytic approach to study the asymptotic dynamics of rough diﬀerential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons’ theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for suﬃciently small noise intensity.