Ultra-slow-roll inflation with quantum diffusion
  We consider the effect of quantum diffusion on the dynamics of the inflaton
during a period of ultra-slow-roll inflation. We extend the
stochastic-$\delta\mathcal{N}$ formalism to the ultra-slow-roll regime and show
how this system can be solved analytically in both the classical-drift and
quantum-diffusion dominated limits. By deriving the characteristic function, we
are able to construct the full probability distribution function for the
primordial density field. In the diffusion-dominated limit, we recover an
exponential tail for the probability distribution, as found previously in
slow-roll inflation. To complement these analytical techniques, we present
numerical results found both by very large numbers of simulations of the
Langevin equations, and through a new, more efficient approach based on
iterative Volterra integrals. We illustrate these techniques with two examples
of potentials that exhibit an ultra-slow-roll phase leading to the possible
production of primordial black holes.
