Abstract
We consider the weakly asymmetric simple exclusion process on the discrete space {1,⋯,n-1}(n∈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 tn 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 to exhibit different boundary behavior than the Cole–Hopf solution.
Highlights
A vast amount of physical phenomena were first described at the macroscopic scale, in terms of the classical partial differential equations (PDEs) of mathematical physics
The macroscopic laws that can arise from microscopic systems can either be partial differential equations (PDEs) or stochastic PDEs (SPDEs) depending on whether one is looking at the convergence to the mean or at the fluctuations around that mean
Among the classical SPDEs is the Kardar–Parisi–Zhang (KPZ) equation which has been first introduced more than thirty years ago in [KPZ86] as the universal law describing the fluctuations of randomly growing interfaces of one-dimensional stochastic dynamics close to a stationary state
Summary
A vast amount of physical phenomena were first described at the macroscopic scale, in terms of the classical partial differential equations (PDEs) of mathematical physics. Among the classical SPDEs is the Kardar–Parisi–Zhang (KPZ) equation which has been first introduced more than thirty years ago in [KPZ86] as the universal law describing the fluctuations of randomly growing interfaces of one-dimensional stochastic dynamics close to a stationary state (as for example, models of bacterial growth, or fire propagation) Since it has generated an intense research activity among the physics and mathematics community. This strategy has been used in more sophisticated models, see [CS18, CST18, CT17, DT16], the applicability of the microscopic Cole–Hopf transformation is limited to a very specific class of particle systems Another way to look at the KPZ equation is via the stochastic Burgers equation (SBE), which is obtained from (1.1) by taking its derivative: if Yt = ∇Zt, Yt satisfies (1.2). None of the models mentioned above admit a microscopic Cole–Hopf transformation, which prevents the use of the methods of [BG97], and in many cases they do not have the structure of a semilinear SPDE, which means that the pathwise approach of [Hai, GIP15] does not apply either
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