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. For concreteness, we apply this approach to stochastic/thermal tunneling for DBI inflation. We show that tunneling rates can be significantly enhanced relative to the Hawking-Moss instanton result due to relativistic DBI effects.