Lecture notes on flow equation approach to singular stochastic PDEs
Paweł Duch
TL;DR
The notes present a continuous-scale renormalization group framework for singular SPDEs driven by white noise, harnessing a flow equation for the scale-dependent effective force to tame ultraviolet divergences. By decomposing the Green function and constructing a scale-dependent force F_{κ,μ}, the approach yields an enhanced-noise representation governed by cumulants that satisfy a Polchinski-type flow; counterterms are fixed by renormalization conditions, and a fixed-point map recovers the original equation in the κ→0 limit. Uniform bounds on cumulants and robust probabilistic estimates enable a well-posed limiting theory across the full subcritical regime, including fractional Laplacians, with applications tying the flow-equation viewpoint to established frameworks like regularity structures and BPHZ. The framework thereby provides a flexible, scale-aware method to construct renormalized solutions and obtain pathwise regularity for a broad class of singular SPDEs, extending beyond Da Prato–Debussche and accommodating non-polynomial nonlinearities. Overall, the work advances a constructive RG methodology for singular SPDEs that unifies probabilistic and analytic techniques and yields quantitative control on renormalization and stochastic fluctuations.
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.