Existence and properties of probability densities for nonlinear Gaussian functionals can be obtained using the stochastic calculus of variations which in Malliavin’s words, proceeds from a merging of differential calculus and probability theory.

In the first part of the lecture, we will give a guided tour through the motivations and the essentials of this calculus and its connections with PDEs. In the second part, we will present some interesting applications, focusing on the study of the probability that sample paths of solutions of stochastic partial differential equations visit deterministic sets.  

We will strive to be accessible to a broad mathematical audience.
