In this paper we focus on the open symmetric exclusion process with parameter $m$ (open SEP($m/2$)), which allows $m$ particles each site and has an open boundary. We generalize the result about hydrodynamic limit for the open SEP$(m/2)$ originally raised in Theorem 4.12 of . We prove that the hydrodynamic limit of the density profile for a $d$-dimensional open SEP$(m/2)$ solves the $(d+1)$-dimensional heat equation with certain initial condition and boundary condition.
"Hydrodynamic limit for a $d$-dimensional open symmetric exclusion process." Electron. Commun. Probab. 25 1 - 8, 2020. https://doi.org/10.1214/20-ECP350