  During inflation explicit perturbative computations of quantum field theories
which contain massless, non-conformal fields exhibit secular effects that grow
as powers of the logarithm of the inflationary scale factor.
Starobinski\u{\i}'s technique of stochastic inflation not only reproduces the
leading infrared logarithms at each order in perturbation theory, it can
sometimes be summed to reveal what happens when inflation has proceeded so long
that the large logarithms overwhelm even very small coupling constants. It is
thus a cosmological analogue of what the renormalization group does for the
ultraviolet logarithms of quantum field theory, and generalizing this technique
to quantum gravity is a problem of great importance. There are two significant
differences between gravity and the scalar models for which stochastic
formulations have so far been given: derivative interactions and the presence
of constrained fields. We use explicit perturbative computations in two simple
scalar models to infer a set of rules for stochastically formulating theories
with these features.
