Exact solution of time-reversed stochastic inflation in the quantum well yields curvature perturbation distributions with faster-decaying exponential tails than forward stochastic inflation.
Tomberg,Numerical stochastic inflation constrained by frozen noise,JCAP04(2023) 042, [2210.17441]
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Spectral solution of the Fokker-Planck operator for hilltop constant-roll inflation shows rare crossing trajectories dominate the mean, so the median yields a coarse-grained ΔN distribution whose exponential tail flattens into a peak near maximal value.
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