Nigel Cutland: An infinitesimal introduction to DEs driven by rough paths

This talk is a beginner's effort to use infinitesimals to understand Lyons rough path integrals and the solution of DEs driven by rough paths.  The intuition underlying Lyons theory is that to unambiguously define an integral  dy(t)=f(x(t))dx(t) when x is a single rough path (such as a typical path of Brownian motion) you have to know some of the iterated integrals of  x against itself. In that sense a rough path is a package consisting of the path itself (in the conventional sense) along with a specification of certain of its iterated integrals.

Infinitesimals provide a way to make the intuition precise and provide a rigorous definition of a Lyons rough path integral. This gives an elementary construction of solutions to DEs driven by rough paths.