Random attractors for dissipative systems with rough noises

Luu Hoang Duc

doi:10.3934/dcds.2021176

Article Contents

2022, Volume 42, Issue 4: 1873-1902. Doi: 10.3934/dcds.2021176

This issue Previous Article Sublacunary sets and interpolation sets for nilsequences Next Article Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

Random attractors for dissipative systems with rough noises

Luu Hoang Duc^1,2, ,

1.
Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
2.
Institute of Mathematics, Vietnam Academy of Science and Technology, 10307 Hanoi, Vietnam

Received: December 2020

Revised: September 2021

Early access: November 2021

Published: April 2022

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Abstract

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.

Keywords:

Stochastic differential equations (SDE),
rough path theory,
rough integrals,
random dynamical systems,
random attractors,
stochastic perturbation,
stochastic stability.

Mathematics Subject Classification: Primary: 37HXX, 60GXX, 60HXX; Secondary: 37H30, 60G22, 60G40, 60H10.

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