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

Open ASEP in the Weakly Asymmetric Regime