  Axions are among the best motivated dark matter candidates. Their production
in the early Universe by the vacuum misalignment mechanism gives rise to
isocurvature perturbations, which are constrained by cosmic microwave
background measurements. In this paper, we compute the axion isocurvature power
spectrum using spectral expansion in the stochastic Starobinsky-Yokoyama
formalism, which captures non-linear effects in the axion dynamics. In contrast
to most of the existing literature, we focus on high inflationary Hubble rates
of order $10^{13}~{\rm GeV}$, and demonstrate that there is a significant
window in which axions can account for all or part of the dark matter abundance
without violating the isocurvature bounds or tensor mode bounds. Crucially, we
find that the isocurvature spectrum is dominated by non-perturbative
contributions in a large part of this window. Therefore the commonly used
linear approximation is not reliable in this region, making the stochastic
approach essential.
