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

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