  We present a new, recursive approach to stochastic inflation which is
self-consistent and solves multiple problems which plagued a certain number of
previous studies, in particular in realistic contexts where the background
spacetime is taken to be dynamical, where there is more than one field present,
especially with a mass hierarchy, or where the role played by back-reaction is
suspected to be important. We first review the formalism of stochastic
inflation as it is usually heuristically presented, that is, deriving the
Langevin equations from the field equations of motion, and summarize previous
results on the subject. We demonstrate where inconsistent approximations to the
Langevin equations are commonly made, and show how these can be avoided. This
set up shares many similarities with quantum Brownian motion and
out-of-equilibrium statistical quantum dynamics. We hence review how path
integral techniques can be applied to the stochastic inflationary context. We
show this formalism to be consistent with the standard approach, develop a
natural perturbative expansion, and use it to calculate the one-loop corrected
Langevin equations.
