A stochastic measure for eternal inflation postprint

Abstract: We use the爏tochastic燼pproach to investigate燼爉easure爁or爏low roll爀ternal爄nflation. The probability爁or爐he universe to have燼爂iven Hubble radius can be calculated in this framework. In燼爏olvable model, it is shown that the probability爁or爐he universe to evolve from燼爏tate with燼爏maller Hubble radius to one with燼爈arger Hubble radius is dominated by the classical probability without the爏tochastic爏ource, while the probability爁or爐he universe to evolve from燼爏tate with燼爈arger Hubble radius to one with燼爏maller one is suppressed by exp(-Delta S), where the de Sitter entropy S arises naturally in this爏tochasticapproach.