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 $\epsilon_1$ if and only if $\epsilon_2^2\ll 6/\epsilon_1$,
with the notable exception of slow-roll. This is due to the failure of the
stochastic $\Delta N$ 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.
