Renormalization: a BPHZ theorem for stochastic PDEs

We show how some of the ideas explored in the previous two lectures can be applied to the study of singular stochastic PDEs.  In particular, we show that although it is not clear a priori what it means to "solve" any particular one of these equations, they can be divided into finite-dimensional families of solutions that are themselves defined in a canonical way.  This is a consequence of a version of the celebrated "BPHZ theorem" which governs renormalisation techniques in perturbative QFT.