The KPZ Limit of ASEP with Boundary

It was recently proved in Corwin and Shen (CPAM, [ CS16 ]) that under weakly asymmetric scaling, the height functions for ASEP with sources and sinks converges to the Hopf–Cole solution of the KPZ equation with inhomogeneous Neumann boundary conditions. In their assumptions [ CS16 ] chose positive values for the Neumann boundary condition, and they assumed initial data which is close to stationarity. By developing more extensive heat-kernel estimates, we clarify and extend their results to negative values of the Neumann boundary parameters, and we also show how to generalize their results to empty initial data (which is very far from stationarity). Combining our result with Barraquand et al. (Duke Math J, [ BBCW17 ]), we obtain the Laplace transform of the one-point distribution for half-line KPZ, and use this to confirm t 1/3 -scale GOE Tracy–Widom long-time fluctuations at the origin.