  During inflation, vacuum quantum fluctuations are amplified and stretched to
astrophysical distances. They give rise to fluctuations in the cosmic microwave
background (CMB) temperature and polarisation, and to large-scale structures in
our universe. They can also trigger the formation of primordial black holes
(PBHs). Such objects could provide the progenitors of the recently detected
black-hole mergers, and constitute part or all of the dark matter. Their
observation would give invaluable access to parts of the inflationary sector
that are unconstrained by the CMB. Since PBHs require large inhomogeneities to
form, they are produced in scenarios where quantum fluctuations substantially
modify the dynamics of the universe. In this habilitation thesis, this
"backreaction" effect is investigated using the stochastic inflation formalism,
an effective theory for the long-wavelengths of quantum fields during
inflation, which can be described in a classical but stochastic way once the
small wavelengths have been integrated out. It describes an inflating
background that gets randomly corrected by the vacuum quantum fluctuations as
they get stretched to large distances. After a brief review of the stochastic
inflation formalism, we explain how it can be combined with standard techniques
of cosmological perturbation theory (the $\delta N$ formalism) to provide the
full probability density function of curvature perturbations in the presence of
non-perturbative quantum diffusion. These results are then applied to PBHs,
where we show that quantum diffusion can change the expected abundance by
several orders of magnitude. Finally, since inflationary models giving rise to
cosmologically relevant PBHs often feature violations of slow roll, the
stochastic-$\delta N$ formalism is generalised to non slow-roll dynamics. We
conclude by highlighting several research directions that remain to be
explored.
