Random attractors for rough differential/evolution equations

We provide an analytic approach to study the long term behavior of rough differential/evolution equations, with the driving noises of Hoelder continuity. Such systems can be solved either with Lyons’ theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths, or with fractional calculus. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of a global pullback attractor.