A BPHZ theorem for stochastic PDEs

A classical result obtained in the 50’s and 60’s by Bogoliubov, Parasiuk, Hepp and Zimmerman provides a prescription on how to renormalise amplitudes of Feynman diagrams arising in perturbative quantum field theory in a consistent way. We will discuss an analogue of this theorem which has both an analytic and a probabilistic interpretation. In particular, we will see that it implies that the solutions to a large class of nonlinear stochastic PDEs depend on their driving noise in a surprisingly rigid way. This rigidity is a mathematical manifestation of the “universality” taken for granted when building our intuition on the large-scale behaviour of probabilistic models.