Malliavin Calculus, Stein's Method and regularity structures

This thesis gives an overview of Stein's method, Malliavin calculus, and regularity structures, and then studies their interaction. After presenting the basic theory and machinery of Malliavin calculus and Stein's method, the Malliavin–Stein method is introduced with an application to the quadratic variation of fractional Brownian motion. We then explain the framework of regularity structures and its use for singular stochastic PDEs. Finally, following Cannizzaro–Friz–Gassiat, Malliavin calculus is combined with regularity structures to show that the solution of the two-dimensional generalized parabolic Anderson model admits a density with respect to the Lebesgue measure.