I will begin by introducing the stochastic formalism for inflation. I will give some motivation for using the stochastic approach and then explain why we may want to consider situations that violate slow roll. I will explain the requirements for stochastic inflation to be a valid approach and discuss whether they are still appropriate when we are away from slow roll. The two requirements that I will consider in detail are the separate universe approach and the choice of gauge that our stochastic equations are written in. I will show that the stochastic formalism is safe to be used, even when slow roll is violated.
arXiv number: 1905.06300 (https://arxiv.org/pdf/1905.06300.pdf)