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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 ofinflation in a quartic potential, we investigate how the inflaton evolves andhow inflation typically ends both with and without volume weighting. Perhapsunexpectedly, complex histories sometimes emerge with volume weighting. Thereward for this excursion into the complex plane is an insight into howvolume-weighted inflation both loses memory of initial conditions and ends viaslow-roll. The slow-roll end of inflation mitigates certain "YoungnessParadox"-type criticisms of the volume-weighted paradigm. Thus it is perhapstime to rehabilitate proper time volume weighting as a viable measure foranswering at least some interesting cosmological questions.

Path Integral for Stochastic Inflation: Non-Perturbative Volume Weighting, Complex Histories, Initial Conditions and the End of Inflation