Stochastic Inflation at NNLO
  Stochastic Inflation is an important framework for understanding the physics
of de Sitter space and the phenomenology of inflation. In the leading
approximation, this approach results in a Fokker-Planck equation that
calculates the probability distribution for a light scalar field as a function
of time. Despite its successes, the quantum field theoretic origins and the
range of validity for this equation have remained elusive, and establishing a
formalism to systematically incorporate higher order effects has been an area
of active study. In this paper, we calculate the next-to-next-to-leading order
(NNLO) corrections to Stochastic Inflation using Soft de Sitter Effective
Theory (SdSET). In this effective description, Stochastic Inflation manifests
as the renormalization group evolution of composite operators. The leading
impact of non-Gaussian quantum fluctuations appears at NNLO, which is presented
here for the first time; we derive the coefficient of this term from a two-loop
anomalous dimension calculation within SdSET. We solve the resulting equation
to determine the NNLO equilibrium distribution and the low-lying relaxation
eigenvalues. In the process, we must match the UV theory onto SdSET at one-loop
order, which provides a non-trivial confirmation that the separation into
Wilson-coefficient corrections and contributions to initial conditions persists
beyond tree level. Furthermore, these results illustrate how the naive
factorization of time and momentum integrals in SdSET no longer holds in the
presence of logarithmic divergences. It is these effects that ultimately give
rise to the renormalization group flow that yields Stochastic Inflation.
