A Stochastic Measure for Eternal Inflation

We use the stochastic approach to investigate the measure for slow roll eternal inflation. The probability for the universe of a given Hubble radius can be calculated in this framework. In a solvable model, it is shown that the probability for the universe to evolve from a state with a smaller Hubble radius to that of a larger Hubble radius is dominated by the classical probability without the stochastic source. While the probability for the universe to evolve from a larger Hubble radius to a smaller one is suppressed by $\exp (-\mathrm{\Delta }S)$, where the de Sitter entropy $S$ arises naturally in this stochastic approach.