Scaling of the Sasamoto-Spohn model in equilibrium

Abstract

We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of [9] and does not use any spectral gap argument.√ 2π e −x 2 /2 dx (see Section 3).AMS 2000 subject classifications. Primary 60K35 secondary 82B20, 60H15 .

Stationary directed polymers and energy solutions of the Burgers equation

“…Uniqueness for this weak formulation on the whole line was proved in [23]. Since [17], this approach was successfully applied to show the convergence of many discrete models to the KPZ/ Burgers equation [18,19,14,32]. One substantial advantage is that it requires very weak quantitative estimates.…”

Scaling limit of stationary coupled Sasamoto-Spohn models

“…The theory of energy solutions has been extremely successful to show convergence of stationary discrete models to the Burgers equation. See for instance [14,13,6,21,22].…”

“…However, this precise definition seems to be the simplest one such that the product of Gaussians is invariant. Convergence to the stochastic Burgers equation was obtained in [18] and [21] in the weakly-asymmetric regime i.e. scaling time by n 2 , space by n and…”

“…This technique originated in [14] where energy solutions for the one-dimensional Burgers equation were introduced. Our approach to the proof of this result is inspired by [15], [21] and [22], and avoids the use of spectral gap estimates. However, we adopt a slightly different though equivalent path to construct the quadratic term in the continuum which allows us to give a much simpler proof.…”

Equilibrium fluctuations for totally asymmetric interacting particle systems

“…As to stationary case, [1] is a celebrating result, which proved that density fluctuation of simple exclusion processes with weak asymmetric jump rates converges to the Cole-Hopf solution of SBE. After that, [6] generalized the result of [1] to wider class of jump rates and remarkably they established a robust way to derive KPZ equation without using Cole-Hopf transformation: [7] for interacting particle systems containing zero-range processes, [5] for a system of stochastic differential equations and [15] for the Sasamoto-Spohn model, which is originally introduced in [21]. Other important class from which KPZ equation is derived is directed polymers, which is introduced in [13] and mathematically analyzed in [14] for the first time.…”