Failure of the stochastic approach to inflation beyond slow-roll

After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter ${ϵ}_1$ if and only if ${ϵ}_2^2\ll 6\mathrm{/}{ϵ}_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $\mathrm{\Delta }N$ formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.