Stochastic inflation with an extremely large number of $e$-folds
  We propose a class of single-field, slow-roll inflation models in which a
typical number of $e$-folds can be extremely large. The key point is to
introduce a very shallow local minimum near the top of the potential in a
hilltop inflation model. In particular, a typical number of $e$-folds is
enhanced if classical behavior dominates around the local minimum such that the
inflaton probability distribution is drifted to the local minimum as a whole.
After the inflaton escapes from the local minimum due to the stochastic
dynamics, the ordinary slow-roll inflation follows and it can generate the
primordial density perturbation consistent with observation. Interestingly, our
scenario inherits the advantages of the old and new inflation: the typical
$e$-folds can be extremely large as in the old inflation, and slow-roll
inflation naturally follows after the stochastic regime as in the new
inflation. In our numerical example, the typical number of $e$-folds can be as
large as $10^{10^{10}}$, which is large enough for various light scalars such
the QCD axion to reach the Bunch-Davies distribution.
