Tóm tắt: In this talk, we present our recent work on the asymptotic dynamics of rough differential equations, with the driving noises of Hoelder 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, while the drift term is locally Lipschitz continuous and of linear growth (which cover many interesting examples).
The first part of the talk aims to construct the framework of random dynamical systems and random attractors, in order to prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded and linear noises. Moreover, if the drift is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity.
The second part focuses on the explicit Euler scheme to approximate the solutions of rough differential equations. Such a numeric scheme converges as the step size goes to zero, under the assumption of bounded or linear noise. In addition, if the drift term is dissipative and of linear growth, then the numeric solutions under constant time steps also generate random dynamical systems which admit random pullback attractors. We also prove that the numeric attractors converge to the continuous attractors as the time step goes to zero.

References:

1. Luu Hoang Duc. Random attractors for dissipative systems with rough noises. to appear in Discrete and Continuous Dynamical Systems Series A.
2. Luu Hoang Duc & Peter Kloeden. Numeric attractors for rough differential equations. In preparation.