  We calculate the full probability density function (PDF) of inflationary
curvature perturbations, even in the presence of large quantum backreaction.
Making use of the stochastic-$\delta N$ formalism, two complementary methods
are developed, one based on solving an ordinary differential equation for the
characteristic function of the PDF, and the other based on solving a heat
equation for the PDF directly. In the classical limit where quantum diffusion
is small, we develop an expansion scheme that not only recovers the standard
Gaussian PDF at leading order, but also allows us to calculate the first
non-Gaussian corrections to the usual result. In the opposite limit where
quantum diffusion is large, we find that the PDF is given by an elliptic theta
function, which is fully characterised by the ratio between the squared width
and height (in Planck mass units) of the region where stochastic effects
dominate. We then apply these results to the calculation of the mass fraction
of primordial black holes from inflation, and show that no more than $\sim 1$
$e$-fold can be spent in regions of the potential dominated by quantum
diffusion. We explain how this requirement constrains inflationary potentials
with two examples.
