KPZ equation from ASEP plus general speed-change drift

We derive the KPZ equation as a continuum limit of height functions in asymmetric simple exclusion processes with a hyperbolic-scale drift that depends on the local particle configuration. To our knowledge, it is a first such result for a general class of particle systems with neither duality nor explicit invariant measures. The new tools to handle the lack of an invariant measure are estimates for Kolmogorov equations that produce a more robust proof of the Kipnis-Varadhan inequality. These tools are not exclusive to KPZ.