Crossover to the stochastic Burgers equation for the WASEP with a slow bond

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho \in (0,1)$. The rate of passage of particles to the right (resp. left) is $\frac{1}{\phantom{n^{\beta }}2}+\frac{a}{2n^{\phantom{\beta }\gamma }}$ (resp. $\frac{1}{\phantom{n^{\beta }}2}-\frac{a}{2n^{\phantom{\beta }\gamma }}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha }{2n^{\beta }}+\frac{a}{2n^{\phantom{\beta }\gamma }}$ (resp. $\frac{\alpha }{2n^{\beta }}-\frac{a}{2n^{\phantom{\beta }\gamma }}$). Above, $\alpha >0$, $\gamma \ge \beta \ge 0$, $a\ge 0$. For $\beta <1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma >\frac{1}{2}$, while for $\gamma =\frac{1}{2}$ it is an energy solution of the stochastic Burgers equation. For $\gamma \ge \beta =1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin’s boundary conditions. For $\gamma \ge \beta >1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann’s boundary conditions.