Langevin Analysis of Eternal Inflation
  It has been widely claimed that inflation is generically eternal to the future, even in models where the inflaton potential monotonically increases away from its minimum. The idea is that quantum fluctuations allow the field to jump uphill, thereby continually revitalizing the inflationary process in some regions. In this paper we investigate a simple model of this process, pertaining to inflation with a quartic potential, in which analytic progress may be made. We calculate several quantities of interest, such as the expected number of inflationary efolds, first without and then with various selection effects. With no additional weighting, the stochastic noise has little impact on the total number of inflationary efoldings even if the inflaton starts with a Planckian energy density. A "rolling" volume factor, i.e. weighting in proportion to the volume at that time, also leads to a monotonically decreasing Hubble constant and hence no eternal inflation. We show how stronger selection effects including a constraint on the initial and final states and weighting with the final volume factor can lead to a picture similar to that usually associated with eternal inflation. 
Steven Gratton and Neil Turok