  If dark matter is made of QCD axions, its abundance is determined by the
vacuum expectation value acquired by the axion field during inflation. The
axion is usually assumed to follow the equilibrium distribution arising from
quantum diffusion during inflation. This leads to the so-called stochastic
window under which the QCD axion can make up all the dark matter. It is
characterised by $10^{10.4}\mathrm{GeV}\leq f\leq 10^{17.2}\mathrm{GeV}$ and
$H_{\mathrm{end}}>10^{-2.2}\mathrm{GeV}$, where $f$ is the axion decay constant
and $H_{\mathrm{end}}$ is the Hubble expansion rate at the end of inflation.
However, in realistic inflationary potentials, we show that the axion never
reaches the equilibrium distribution at the end of inflation. This is because
the relaxation time of the axion is much larger than the typical time scale
over which $H$ varies during inflation. As a consequence, the axion acquires a
quasi-flat distribution as long as it remains light during inflation. This
leads us to reassessing the stochastic axion window, and we find that $
10^{10.3}\mathrm{GeV}\leq f\leq 10^{14.1}\mathrm{GeV}$ and
$H_{\mathrm{end}}>10^{-13.8}\mathrm{GeV}$.
