A recent breakthrough in the area of stochastic PDEs was the development of Martin Hairer’stheory of Regularity Structures. Martin Hairer used this theory to show how one can navigate around ill-defined products of (Schwartz) distributions that arise in stochastic PDEs. This work was later recognized and lead to Martin Hairer being awarded the Fields Medal in 2014.
In this talk, I will present the main inspiration for regularity structures, which is the theory of rough paths. I will focus on how one defines the following ’integral’
where and are Hölder continuous of order less than .