Random walks in random environment as rough paths

Invariance principles à la Donsker are well understood for many random walks in random environments. But if we encounter the random walk as noise acting on a differential equation, then we have to study the invariance principle in a fine topology in order to understand the convergence properties of the solutions: The so called rough path topology. In my talk I will briefly present this topology and then discuss a general invariance principle that describes the fluctuations in the ergodic theorem for stationary Markov processes in rough path topology. As an application, we will see a rough path invariance principle for the random conductance model (a particular class of random walks in random environments). The talk is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.