Abstract

We prove universality of a macroscopic behavior of solutions of a large class of semilinear parabolic SPDEs on ${ℝ}_{+}\times \mathbb{𝕋}$ with fractional Laplacian ${(-\mathrm{\Delta })}^{\sigma ∕2}$, additive noise and polynomial nonlinearity, where $\mathbb{𝕋}$ is the $d$-dimensional torus. We consider the weakly nonlinear 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 nonlinearity our analysis covers the whole subcritical regime $\sigma >2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.

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