Crossover to the stochastic Burgers equation for the WASEP with a slow bond (Q328478)

The authors perturb the dynamics of the symmetric simple exclusion process by a weak asymmetry of strength \(a n^{2-\gamma}\) (with \(a\geq 0\) and \(\gamma\geq \frac{1}{2}\)) and the introduction of a slow jump rate at the bond \(\{-1,0\}\) where the rate to the right (left, respectively) becomes \(\frac{\alpha}{2 n^{\beta}}+ \frac{a}{2 n^{\gamma}}\) (\(\frac{\alpha}{2 n^{\beta}}- \frac{a}{2 n^{\gamma}}\), respectively) with \(\alpha>0\) and \(\gamma\geq \beta\). The system is taken under the invariant state, namely a Bernoulli product measure. In this framework, they first give a new version of the second-order Boltzmann-Gibbs principle, which allows them to replace certain additive functionals of local functions of the dynamics by additive functionals given in terms of the density of particles. This is obtained through new arguments avoiding the usual spectral gap inequality. Then they show that for \(\beta<1\) the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space for \(\gamma>\frac{1}{2}\), while for \(\gamma =\frac{1}{2}\) it is an energy solution of the stochastic Burgers equation. This field is an Ornstein-Uhlenbeck process associated to the heat equation with Robin boundary conditions for \(\beta=1\) and Neumann boundary conditions for \(\beta >1\).