Mathematics

Dr. Atul Shekhar
ISI Bangalore

Rough Path theory was introduced by T. Lyons in order to make sense of calculus based on paths which are very irregular and classical theory fails to apply, for example, Brownian motion. A pathwise notion of integration, called rough integration, will be defined and it will be shown that it matches with classical integration theories. The theory applies to many other (non-martingale) stochastic processes, e.g., fractional Brownian motion. Such pathwise understanding comes with many applications such as stability results for SDE under smooth approximations of noise.

In a recent work by M. Hairer, idea of rough path theory was to used to give natural solution concepts to many degenerate SPDEs like the Kardar-Parisi-Zhang (KPZ) equation.

This mini course intends to give a mild introduction to the subject.