Stochastic Dynamics and Eternal Inflation

In describing eternal inflation, significant progress can be made within the stochastic approach, based on the fact that local evolution includes the influence of quantum fluctuations, which are introduced through the Langevin equation, from which, in the next step, the Fokker-Planck-Kolmogorov equation is derived. We discuss the details of this derivation, as well as the problem of the absence of stationary distributions. Using the connection with the Abel equation of the first kind, established in our earlier work, we show that there is a natural generalization of this equation beyond the slow-roll approximation, and the resulting equation admits a stationary distribution.