Rough walks

Random walks in random environment (RWRE) have been extensively studied in the last half-century. Two prototypical cases are the reversible and the ballistic classes and even though they are fundamentally different, functional central limit theorems (FCLT) are known to hold in both. This is done using rather different techniques; Kipnis-Varadhan's theory for additive functional of Markov processes is applicable in the reversible case while for the ballistic class the main feature is a regenerative structure which guarantees that the process contains a random walk as a subsequence. 

Rough path theory is a deterministic theory of integration which allows path integration with respect to singular signals in a continuous manner. It typically provides a framework to solve stochastic differential equations (ordinary and partial) driven by a singular noise.    

We shall discuss some recent contributions, in which we lift additive functionals of Markov processes and regenerative processes to the rough path space and obtain an enhanced FCLT.

The first level of the limiting rough path is naturally the Brownian motion but, somewhat surprisingly a deterministic linear perturbation appears in the second level. We characterize the latter in various ways. Except for the immediate application to SDE, this provides some new information on the structure of the limiting path. The aforementioned classes of RWRE are covered as special cases.  

Based on collaborations with Johaness Bäumler, Noam Berger, Jean-Dominique Deuschel, Olga Lopusanschi, Martin Slowik and Nicolas Perkowski.