  Particle theorists typically use expectation values to study the quantum
back-reaction on inflation, whereas many cosmologists stress the stochastic
nature of the process. While expectation values certainly give misleading
results for some things, such as the stress tensor, we argue that operators
exist for which there is no essential problem. We quantify this by examining
the stochastic properties of a noninteracting, massless, minimally coupled
scalar on a locally de Sitter background. The square of the stochastic
realization of this field seems to provide an example of great relevance for
which expectation values are not misleading. We also examine the frequently
expressed concern that significant back-reaction from expectation values
necessarily implies large stochastic fluctuations between nearby spatial
points. Rather than viewing the stochastic formalism in opposition to
expectation values, we argue that it provides a marvelously simple way of
capturing the leading infrared logarithm corrections to the latter, as
advocated by Starobinsky.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory