Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes

We prove here that the height function associated to non-simple exclusion processes with arbitrary jump-length converges to the solution of the Kardar-Parisi-Zhang SPDE under suitable scaling and renormalization. This extends the work of Dembo and Tsai (Commun Math Phys 341(1):219-261, 2016) for arbitrary jump-length and Goncalves and Jara (Stoch Process Appl 127(12):4029-4052, 2017) for the non-stationary regime. Thus we answer a "Big Picture Question " from the AIM workshop on KPZ and also expand on the almost empty set of non-integrable and non-stationary particle systems for which weak KPZ universality is proven. We use an approximate microscopic Cole-Hopf transform like in Dembo and Tsai (2016) but we develop tools to analyze local statistics of the particle system via local equilibrium and work of Goncalves and Jara (2017). Local equilibrium is done via the one-block step in Guo et al. (Commun Math Phys 118:31, 1988) for path-space/dynamic statistics.