Flow equation approach to singular stochastic PDEs

TL;DR

This paper establishes the universality of macroscopic behavior for a broad class of singular semi-linear parabolic SPDEs with fractional Laplacian, using a novel Wilsonian renormalization group approach, especially covering the sub-critical regime in four dimensions.

Contribution

It introduces a new solution theory for singular SPDEs via the Polchinski flow equation, avoiding complex algebraic and combinatorial methods.

Findings

Proves the existence of a universal macroscopic law for the class of SPDEs studied.

Develops a new renormalization group-based solution framework for singular SPDEs.

Covers the entire sub-critical regime for $d=4$ and cubic non-linearity.

Abstract

We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with fractional Laplacian $(-\Delta)^{\sigma/2}$, additive noise and polynomial non-linearity, where $\mathbb{T}$ is the $d$-dimensional torus. We consider the weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. We prove that the macroscopic scaling limit exists and has a universal law characterized by parameters of the relevant perturbations of the linear equation. We develop a new solution theory for singular SPDEs of the above-mentioned form using the Wilsonian renormalization group theory and the Polchinski flow equation. In particular, in the case of $d=4$ and the cubic non-linearity our analysis covers the whole sub-critical regime $\sigma>2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.