A Course on Rough Paths: With an Introduction to Regularity Structures

Introduction.- The space of rough paths.- Brownian motion as a rough path.- Integration against rough paths.- Stochastic integration and Ito'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.