Stochastic inflation describes the global structure of the inflationary universe
by modeling the super-Hubble dynamics as a system of matter fields coupled to
gravity where the sub-Hubble field fluctuations induce a stochastic force into
the equations of motion. The super-Hubble dynamics are ultralocal, allowing us
to neglect spatial derivatives and treat each Hubble patch as a separate
universe. This provides a natural framework in which to discuss probabilities on
the space of solutions and initial conditions. In this article we derive an
evolution equation for this probability for an arbitrary class of matter
systems, including DBI and k-inflationary models, and discover equilibrium
solutions that satisfy detailed balance. Our results are more general 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 slow roll conditions. We
discuss in general terms the conditions for eternal inflation to set in, and we
give explicit numerical solutions of highly stochastic, quasi-stationary
trajectories in the relativistic DBI regime. Finally, we show that the
probability for stochastic/thermal tunneling can be significantly enhanced
relative to the Hawking-Moss instanton result due to relativistic DBI effects.