Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation: implications for primordial black hole production

We consider quantum diffusion in ultra-slow-roll (USR) inflation. Using the $\mathrm{\Delta }N$ formalism, we present the first stochastic calculation of the probability distribution $P(\mathcal{R})$ of the curvature perturbation during USR. We capture the non-linearity of the system, solving the coupled evolution of the coarse-grained background with random kicks from the short wavelength modes, simultaneously with the mode evolution around the stochastic background. This leads to a non-Markovian process from which we determine the highly non-Gaussian tail of $P(\mathcal{R})$. Studying the production of primordial black holes in a viable model, we find that stochastic effects during USR increase their abundance by a factor $\sim {10}^5$ compared to the Gaussian approximation.