Flow equation approach to singular stochastic PDEs

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 $(-\hat{I}”{)}^{\ddot{I}ƒ/2}$, 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 $\ddot{I}ƒ>2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.

146 pages, minor changes to match the published version, added list of symbols