  Quantum diffusion describes the inflow of vacuum quantum fluctuations as they
get amplified by gravitational instability, and stretched to large distances
during inflation. In this picture, the dynamics of the universe's expansion
becomes stochastic, and the statistics of the curvature perturbation is encoded
in the distribution of the duration of inflation. This provides a
non-perturbative framework to study cosmological fluctuations during inflation,
which is well-suited to the case of primordial black holes since they originate
from large fluctuations. We show that standard, perturbative expectations for
the primordial black hole abundance can be significantly modified by
quantum-diffusion effects, and we identify a few open challenges.
