  In this paper we present a path integral formulation of stochastic inflation,
in which volume weighting can easily be implemented. With an in-depth study of
inflation in a quartic potential, we investigate how the inflaton evolves and
how inflation typically ends both with and without volume weighting. Perhaps
unexpectedly, complex histories sometimes emerge with volume weighting. The
reward for this excursion into the complex plane is an insight into how
volume-weighted inflation both loses memory of initial conditions and ends via
slow-roll. The slow-roll end of inflation mitigates certain "Youngness
Paradox"-type criticisms of the volume-weighted paradigm. Thus it is perhaps
time to rehabilitate proper time volume weighting as a viable measure for
answering at least some interesting cosmological questions.
