$\Delta\mathcal{N}$ and the stochastic conveyor belt of Ultra Slow-Roll

We analyse field fluctuations during an Ultra Slow-Roll phase in the stochastic picture of inflation and the resulting non-Gaussian curvature perturbation, fully including the gravitational backreaction of the field's velocity. By working to leading order in a gradient expansion, we first demonstrate that consistency with the momentum constraint of General Relativity prevents the field velocity from having a stochastic source, reflecting the existence of a single scalar dynamical degree of freedom on long wavelengths. We then focus on a completely level potential surface, $V=V_0$, extending from a specified exit point $\phi_{\rm e}$, where slow roll resumes or inflation ends, to $\phi\rightarrow +\infty$. We compute the probability distribution in the number of e-folds $\mathcal{N}$ required to reach $\phi_{\rm e}$ which allows for the computation of the curvature perturbation. We find that, if the field's initial velocity is high enough, all points eventually exit through $\phi_{\rm e}$ and a finite curvature perturbation is generated. On the contrary, if the initial velocity is low, some points enter an eternally inflating regime despite the existence of $\phi_{\rm e}$. In that case the probability distribution for $\mathcal{N}$, although normalizable, does not possess finite moments, leading to a divergent curvature perturbation.