  We study the inflationary evolution of a scalar field $h$ with an unstable
potential for the case where the Hubble parameter $H$ during inflation is
larger than the instability scale $\Lambda_I$ of the potential. Quantum
fluctuations in the field of size $\delta h \sim \frac{H}{2 \pi}$ imply that
the unstable part of the potential is sampled during inflation. We investigate
the evolution of these fluctuations to the unstable regime, and in particular
whether they generate cosmological defects or even terminate inflation. We
apply the results of a toy scalar model to the case of the Standard Model (SM)
Higgs boson, whose quartic evolves to negative values at high scales, and
extend previous analyses of Higgs dynamics during inflation utilizing
statistical methods to a perturbative and fully gauge-invariant formulation. We
show that the dynamics are controlled by the renormalization group-improved
quartic coupling $\lambda(\mu)$ evaluated at a scale $\mu = H$, such that Higgs
fluctuations are enhanced by the instability if $H > \Lambda_I$. Even if $H >
\Lambda_I$, the instability in the SM Higgs potential does not end inflation;
instead the universe slowly sloughs off crunching patches of space that never
come to dominate the evolution. As inflation proceeds past 50 $e$-folds, a
significant proportion of patches exit inflation in the unstable vacuum, and as
much as 1% of the spacetime can rapidly evolve to a defect. Depending on the
nature of these defects, however, the resulting universe could still be
compatible with ours.
