  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.
