Statistics of coarse-grained cosmological fields in stochastic inflation
  We present a generic framework to compute the one-point statistics of
cosmological perturbations, when coarse-grained at an arbitrary scale $R$, in
the presence of quantum diffusion. Making use of the stochastic-$\delta N$
formalism, we show how it can be related to the statistics of the amount of
expansion realised until the scale $R$ crosses out the Hubble radius. This
leads us to explicit formulae for the probability density function (PDF) of the
curvature perturbation, the comoving density contrast, and the compaction
function. We then apply our formalism to the calculation of the mass
distribution of primordial black holes produced in a single-field model
containing a "quantum well" (i.e. an exactly flat region in the potential). We
confirm that the PDFs feature heavy, exponential tails, with an additional
cubic suppression in the case of the curvature perturbation. The large-mass end
of the mass distribution is shown to be mostly driven by
stochastic-contamination effects, which produce black holes more massive than
those naively expected. This work bridges the final gap between the
stochastic-inflation formalism and the calculation of the mass distribution of
astrophysical objects such as primordial black holes, and opens up various
prospects that we finally discuss.
