Lecture notes on flow equation approach to singular stochastic PDEs

TL;DR

This paper introduces a flow equation approach inspired by renormalization group techniques to analyze singular stochastic PDEs, providing a systematic way to understand their behavior across scales.

Contribution

It develops a novel flow equation framework for singular SPDEs, extending the renormalization group method to a broad class of equations with fractional Laplacians.

Findings

The approach applies to the entire subcritical regime of singular SPDEs.

It captures the evolution of nonlinear terms across spatial scales.

The method offers a systematic solution to the renormalization problem.

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.