Inflation is always semi-classical: Diffusion domination overproduces
  Primordial Black Holes
  We use the Hamilton-Jacobi (H-J) formulation of stochastic inflation to
describe the evolution of the inflaton during a period of Ultra-Slow Roll
(USR), taking into account the field's velocity and its gravitational
backreaction. We demonstrate how this formalism allows one to modify existing
slow-roll (SR) formulae to be fully valid outside of the SR regime. We then
compute the mass fraction, $\beta$, of Primordial Black Holes (PBHs) formed by
a plateau in the inflationary potential. By fully accounting for the inflaton
velocity as it enters the plateau, we find that PBHs are generically
overproduced before the inflaton's velocity reaches zero, ruling out a period
of free diffusion or even stochastic noise domination on the inflaton dynamics.
We also examine a local inflection point and similarly conclude that PBHs are
overproduced before entering a quantum diffusion dominated regime. We therefore
surmise that the evolution of the inflaton is always predominantly classical
with diffusion effects always subdominant. Both the plateau and the inflection
point are characterized by a very sharp transition between the under- and
over-production regimes. This can be seen either as severe fine-tunning on the
inflationary production of PBHs, or as a very strong link between the fraction
$\beta$ and the shape of the potential and the plateau's extent.
