Rough paths and random dynamical systems

We aim to study the long time behaviour of solutions to rough differential equations (in the sense of Lyons) driven by a rough paths valued stochastic process. Most equations of interest contain an unbounded drift term. Our first contribution is to extend the existing theory to equations containing these terms. In the second part of the talk, we connect rough paths theory to the theory of random dynamical systems. In particular, we show that rough differential equations naturally induce random dynamical systems provided the driving rough paths valued process has stationary increments. This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).