Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

Abstract

We consider the weakly asymmetric simple exclusion process on the discrete space {1, ..., n − 1} (n ∈ ), in contact with stochastic reservoirs, both with density ρ ∈ (0, 1) at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter ρ. Under time diffusive scaling t n 2 and for ρ = 1 2 , when the asymmetry parameter is taken of order 1/ n, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and different from the Cole-Hopf solution.

Open ASEP in the Weakly Asymmetric Regime

“…By (1.2), we may compute the effective densities A D˛=p and B D ı=q. In the special one-parameter subcase when the effective densities on the left and right are equal, i.e., % A D % B in (1.2), then one may check (see, e.g., [42]) that the invariant measure for open ASEP is a product of Bernoulli random variables with density % A . The equality % A D % B amounts to the relationships between A and B that…”

“…However, in a special one-parameter case (see Remark 2.9) the invariant measure reduces to product Bernoulli. Upon sending this present paper out for comments, we learned that [42] are completing a work in which they develop the energy solutions approach to study the KPZ limit of open ASEP in the special subcase when the invariant measure is product Bernoulli, and the effective density is exactly 1 2 . This is a special point in the two-dimensional family of parameters we consider here.…”

“…would like to acknowledge Kirone Mallick for discussions regarding the matrix product ansatz. We also thank Marielle Simon, Patricia Gonçalves, and Nicolas Perkowski for letting us know about their work in preparation [42] when we sent them a draft of our paper, and in turn for sending and discussing with us their draft. We would also like to thank the referee for the helpful comments.…”

The KPZ Limit of ASEP with Boundary

“…Two weeks ago, there was a related result which was posted by Goncalves, Perkowski, and Simon in [GPS17]. In that paper, they prove (under the assumption of stationary initial data and boundary parameters A = B = − 1 2 ) weakly asymmetric convergence of ASEP to stochastic Burgers using very different methods than the ones we use here (the Cole-Hopf transform).…”

The KPZ equation on the real line

“…However, in many situations (including for the KPZ equation) the higher order terms can only be constructed with the help of a suitable renormalisation, and this means that the solution we eventually find does not solve the original equation, but a renormalised version of it [BCCH17]. But while we now have a good understanding in what sense the Cole-Hopf solution solves the KPZ equation and how to interpret its renormalisation, all this is restricted to the equation on the torus or in a finite volume with boundary conditions [GH18a, CS16,GPS17]. Since one of the main interests in the KPZ equation comes from its large scale behavior it would be more natural to solve it on R, a space that can be arbitrarily rescaled.…”