Open ASEP in the Weakly Asymmetric Regime

We consider ASEP on a bounded interval and on a half line with sources and sinks. On the full line, Bertini and Giacomin proved convergence under weakly asymmetric scaling of the height function to the solution of the KPZ equation. We prove here that under similar weakly asymmetric scaling of the sources and sinks as well, the bounded interval ASEP height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides (different parameter for each side); and likewise for the half line ASEP to KPZ on a half line. This result can be interpreted as showing that the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP.