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 $\frac1{\vphantom{n^\beta}2}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac1{\vphantom{n^\beta}2}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha}{2n^\beta}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$). Above, $\alpha>0$, $\gamma\geq \beta\geq 0$, $a\geq 0$. For $\beta<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma>\frac12$, while for $\gamma = \frac12$ it is an energy solution of the stochastic Burgers equation. For $\gamma\geq\beta=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $\gamma\geq\beta> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.