  It has been suggested that low energy effective field theories should satisfy
given conditions in order to be successfully embedded into string theory. In
the case of a single canonically normalized scalar field this translates into
conditions on its potential and the derivatives thereof. In this Letter we
revisit stochastic models of small field inflation and study the compatibility
of the swampland constraints with entropy considerations. We show that
stochastic inflation either violates entropy bounds or the swampland criterium
on the slope of the scalar field potential. Furthermore, we illustrate that
such models are faced with a graceful exit problem: any patch of space which
exits the region of eternal inflation is either not large enough to explain the
isotropy of the cosmic microwave background, or has a spectrum of fluctuations
with an unacceptably large red tilt.
