Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

Abstract

We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distributions in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson model. MSC: 60H15, 60F05, 30H25

The KPZ equation on the real line

“…In the first section we introduce techniques from paracontrolled calculus for SPDEs in a weighted setting, cf. [GIP15, GP17,MP17]. Among them are the commutation and product estimates from Lemmata 2.10 and 2.8, as well as tailor-made Schauder estimates for the weighted setting, e.g.…”

“…This theory is presented in [Tri10] or [Bjö66]. For a simple and handson introduction to all the tools we need we refer to [MP17]. Consider the function ω(x) = |x| δ , δ ∈ (0, 1) with δ fixed once and for all.…”

Some recent progress in singular stochastic partial differential equations

“…The proof of [CGP17] is based on paracontrolled distributions as we introduced in Section 4.2. The result [CGP17] is further generalized by [MP17] which proves such a convergence result for the nonlinear parabolic Anderson model (3.4) where the factor f (u) models some interaction between the individual particles. The proofs require showing convergence of the perturbative objects discussed in Section 4.2 in the discrete settings, which is one of the main technical challenge -the argument of (3.4) relies on some general tools developed by [CSZ17a].…”

Stationary directed polymers and energy solutions of the Burgers equation

“…This breakthrough was shortly followed by an existence and uniqueness theory for KPZ/Burgers in the framework of paracontrolled distributions [22]. As in the case of regularity structures, this theory can be used to treat other stochastic PDEs beyond KPZ [9] and is amenable to show the convergence of discrete models [10,35]. Furthermore, this theory can be succesfully applied to the KPZ equation on the whole real line [41].…”

“…The works [28,8] provide a general framework to treat discrete models in the context of regularity structures. In the framework of paracontrolled distributions, the work [22] was successful in giving a first proof of the convergence of the periodic Sasamoto-Spohn model and the works [10,35] developed robust arguments to treat stochastic PDEs on the lattice. In the context of energy solutions, speed-change exclusion dynamics was treated in [17,18], interacting diffusions in [14] and the Sasamoto-Spohn model on the whole line in [32].…”