We consider the weakly asymmetric simple exclusion process on the discrete space , in contact with stochastic reservoirs, both with density at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter . Under time diffusive scaling and for , when the asymmetry parameter is taken of order , 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 to exhibit different boundary behavior than the Cole–Hopf solution.
[FGN17] Equilibrium fluctuations for the slow boundary exclusion process, From Particle Systems to Partial Differential Equations (Springer Proceedings in Mathematics & Statistics) Volume 209 (2017), pp. 177-197 | MR | Zbl
[GP16] The Hairer–Quastel universality result at stationarity, RIMS Kôkyûroku Bessatsu, Volume B59 (2016), pp. 101-115 | Zbl
[GP18b] Probabilistic approach to the stochastic Burgers equation, Stochastic partial differential equations and related fields (Springer Proceedings in Mathematics & Statistics) Volume 229 (2018), pp. 515-527 | DOI | MR | Zbl
[LMO08] Stationary and non-equilibrium fluctuations in boundary driven exclusion processes, Markov Process. Relat. Fields, Volume 14 (2008) no. 2, pp. 165-184 | Zbl
[Wal86] An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984 (Lecture Notes in Mathematics) Volume 1180, Springer, 1986, pp. 265-439 | DOI | MR | Zbl