Stochastic inflation with gradient interactions

Stochastic inflation rests on the separate-universe approximation, i.e. the ability to describe long-wavelength fluctuations in an inflating universe as homogeneous perturbations of its background dynamics. Although this approximation is valid in most cases, it has been recently pointed out that it breaks down during transition periods between attractor and non-attractor phases. Such transitions are ubiquitous in single-field models giving rise to enhanced perturbations at small scales, that are required to form primordial black holes. The current inability to apply the stochastic-inflation program in such models is therefore one of the main obstacles to investigating the role of backreaction in primordial-black-hole scenarios. In this work, we show how gradient interactions can be incorporated in stochastic inflation, via a set of Langevin equations of higher dimension. We apply our formalism to a few cases of interest, including one with a sharp transition. In all cases, in the classical limit we show that gradient corrections as predicted from cosmological perturbation theory are properly recovered. We uncover the existence of a "pullback" effect by which the tails of the first-passage-time distributions are dampened by gradient interactions. We finally discuss the role of backreaction in the presence of gradient interactions.