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Stochastic inflation describes the global structure of the inflationaryuniverse by modeling the super-Hubble dynamics as a system of matter fieldscoupled to gravity where the sub-Hubble field fluctuations induce a stochasticforce into the equations of motion. The super-Hubble dynamics are ultralocal,allowing us to neglect spatial derivatives and treat each Hubble patch as aseparate universe. This provides a natural framework in which to discussprobabilities on the space of solutions and initial conditions. In this articlewe derive an evolution equation for this probability for an arbitrary class ofmatter systems, including DBI and k-inflationary models, and discoverequilibrium solutions that satisfy detailed balance. Our results are moregeneral than those derived assuming slow roll or a quasi-de Sitter geometry,and so are directly applicable to models that do not satisfy the usual slowroll conditions. We discuss in general terms the conditions for eternalinflation to set in, and we give explicit numerical solutions of highlystochastic, quasi-stationary trajectories in the relativistic DBI regime.Finally, we show that the probability for stochastic/thermal tunneling can besignificantly enhanced relative to the Hawking-Moss instanton result due torelativistic DBI effects.

Stochastic Inflation Revisited: Non-Slow Roll Statistics and DBI Inflation