Lecture notes on the flow equation approach to singular stochastic PDEs

Abstract

The flow equation approach is a robust framework applicable to a broad class of singular SPDEs, including those with fractional Laplacians, throughout the entire subcritical regime. Inspired by Wilson's renormalization group, this method studies the coarse-grained process, which captures the behaviour of solutions across spatial scales. The corresponding flow equation describes how the nonlinear terms in the effective dynamics evolve with the coarse-graining scale, playing a role analogous to the Polchinski equation in quantum field theory. The renormalization problem is then solved inductively by imposing appropriate boundary conditions on the flow equation.