A Course on Rough Paths

Peter K. Friz is presently a full professor at TU and WIAS Berlin, previous professional affiliations include Cambridge University, Merrill Lynch. He holds a PhD from the Courant Institute of New York University. PKF has made contributions to the understanding of the Navier-Stokes equation as dynamical system, pioneered new asymptotic techniques in financial mathematics and has written many influential papers on the applications of rough path theory to stochastic analysis, ranging from the interplay of rough paths with Malliavin calculus to a (rough-) pathwise view on non-linear SPDEs. Jointly with N. Victoir he authored a monograph on stochastic processes as rough paths. Martin Hairer FRS is currently Regius Professor of Mathematics at the University of Warwick. He has mostly worked in the field of stochastic partial differential equations in particular, and in stochastic analysis and stochastic dynamics in general. He made fundamental advances in various directions such as the study of hypoelliptic and/or hypocoercive diffusions, the development of an ergodic theory for stochastic PDEs, the systematisation of the construction of Lyapunov functions for stochastic systems, the development of a general theory of ergodicity for non-Markovian systems, multiscale analysis techniques, etc. Most recently, he has worked on applying rough path techniques to the analysis of certain ill-posed stochastic PDEs and introduced the theory of regularity structures. Martin Hairer won the Fields Medal 2014.

Introduction.- The space of rough paths.- Brownian motion as a rough path.- Integration against rough paths.- Stochastic integration and Itˆo’s formula.- Doob–Meyer type decomposition for rough paths.- Operations on controlled rough paths.- Solutions to rough differential equations.- Stochastic differential equations.- Gaussian rough paths.- Cameron–Martin regularity and applications.- Stochastic partial differential equations.- Introduction to regularity structures.- Operations on modelled distributions.- Application to the KPZ equation.