# Solution of the Kolmogorov equation for TASEP

## TL;DR

This paper offers a straightforward proof that the TASEP transition probabilities satisfy the Kolmogorov backward equation, also extending the method to the PushASEP particle system, thereby clarifying the mathematical structure of these stochastic processes.

## Contribution

It provides an elementary proof confirming the TASEP transition probabilities solve the Kolmogorov backward equation and extends this approach to PushASEP.

## Key findings

- Confirmed TASEP transition probabilities solve the Kolmogorov backward equation
- Extended the proof technique to PushASEP system
- Simplified understanding of TASEP and PushASEP dynamics

## Abstract

We provide a direct and elementary proof that the formula obtained in [MQR17] 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.