The Kardar–Parisi–Zhang Equation as Scaling Limit of Weakly Asymmetric Interacting Brownian Motions

Abstract

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara [GJ14] and the corresponding uniqueness result of [GP15a].

Energy solutions of KPZ are unique

“…Following [GJ14] it has been shown for a variety of models that their fluctuations subsequentially converge to energy solutions of the KPZ equation or the SBE, for example for zero range processes and kinetically constrained exclusion processes in [GJS15], various exclusion processes in [GS15,FGS16,BGS16,GJ16], interacting Brownian motions in [DGP16], and Hairer-Quastel type SPDEs in [GP16]. This is coherent with the conjecture that the SBE/KPZ equation describes the universal behavior of a wide class of conservative dynamics or interface growth models in the particular limit where the asymmetry is "small" (depending on the spatial scale), the so called weak KPZ universality conjecture, see [Cor12,Qua14,QS15,Spo16].…”

Some recent progress in singular stochastic partial differential equations

“…admits a one parameter family of stationary measures parametrized by λ ∈ R: and Z λ = R e λu−V (u) du is a normalization constant. More precisely, [DGP17] proved the following result. Assuming u 0 (i) = φ 0 (i) − φ 0 (i − 1) has the probability distribution µ λ for a fixed λ ∈ R, and let ρ (λ) := R u p λ (u)du be the mean of the coordinates u(j) under µ λ .…”

Stationary directed polymers and energy solutions of the Burgers equation

“…For instance, it does not allow to obtain direct convergence to Burgers equation for the occupation field of the exclusion process as in [17] although it can be used to show convergence of its height function to the Cole-Hopf solution of KPZ. Second, this approach relies heavily on the availability of a discrete Cole-Hopf transform which is not available for relevant models such as the Sasamoto-Spohn model [42] or the coupled diffusions considered in [14].…”

“…Convergence at fixed times. A straightforward adaptation of the arguments in [14], Section 4.1.1, shows that X n t converges to a white noise for each fixed time t ∈ [0, T ]. This in turns proves that the limit satisfies property (S).…”