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

TL;DR

This work reframes inflationary dynamics through stochastic inflation beyond slow-roll by exploiting an ultralocal gravity limit and general noncanonical kinetic terms. It derives both Langevin and phase-space Fokker-Planck descriptions, identifies a scaling symmetry that yields a conserved curvature perturbation and a detailed-balance equilibrium, and discusses implications for eternal inflation and stochastic tunneling. The framework is specialized to DBI (and k-inflation) models, where the diffusion coefficients and sound speed $c_s$ modify the stochastic dynamics and can enhance Hawking-Moss-type tunneling in relativistic regimes. Numerical results illustrate rich stochastic DBI trajectories, including metastable, ultra-relativistic behavior at large field values, and the analysis clarifies how stochastic effects influence inflationary histories and measure issues.

Abstract

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.