Flow equation approach to singular stochastic PDEs

We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on R₊ with fractional Laplacian (-Δ)σ/2, additive noise and polynomial non-linearity, where 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 the case of $d=4$ and the cubic non-linearity our analysis covers the whole sub-critical regime σ>2. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.