We provide a direct and elementary proof that the formula obtained in (Matetski, Quastel and Remenik (2016)) for the TASEP transition probabilities for general (one-sided) initial data solves the Kolmogorov backward equation. The same method yields the solution for the related PushASEP particle system.
"Solution of the Kolmogorov equation for TASEP." Ann. Probab. 48 (5) 2344 - 2358, September 2020. https://doi.org/10.1214/20-AOP1425