Malliavin Calculus for the nonlinear Parabolic Anderson Model

Many nonlinear stochastic PDES arising in statistical mechanics are ill-posed in the sense that one cannot give a canonical meaning to the nonlinearity. Nevertheless, Martin Hairer’s theory of regularity structures provides us with a good notion of solution for a large class of such equations (KPZ equation,stochastic quantization,...). One of the simplest equations to which this theory can (and should) be applied is the generalized 2D Parabolic Anderson Model : $$(\partial_t - \Delta) u = f(u) \xi, \;\;\; u(0)=u_0,$$ where $\xi$ is a spatial white noise on the torus $\mathbb{T}^2$.

In my talk, after a quick overview of the theory of regularity structures (in the particular case of this equation), I will explain how it can be combined with classical tools of Malliavin calculus, which allows in particular to obtain absolute continuity results for the marginal laws of the solutions. This is based on joint work with G. Cannizzaro and P. Friz (TU Berlin).