Weak universality and singular stochastic PDEs

The macroscopic or mesoscopic dynamics of many systems interacting with a random or chaotic environment can be described in terms of singular (i.e. classically ill-posed) stochastic partial differential equations. Typically, such stochastic PDEs depend only on a few parameters and govern the large-scale behavior of a huge number of different microscopic systems. This property is called universality.

In the talk, I will discuss the proof of the universality of the macroscopic scaling limit of solutions of a class of parabolic stochastic PDEs with fractional Laplacian, additive noise and polynomial non-linearity. I consider the so-called weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. The result applies to situations when the singular stochastic PDE obtained in the scaling limit is close to critical and extends some of the existing universality results about the continuous interface growth models and the phase coexistence models whose large scale behavior is governed by the KPZ equation and the dynamical $\Phi^4_3$ model, respectively.

The proof uses a novel approach to singular stochastic PDEs based on the renormalization group flow equation. A nice feature of the method is that it covers the full sub-critical (i.e. super-renormalizable) regime, does not use any diagrammatic representation and avoids all combinatorial problems. Based on arXiv:2109.11380.