Thermal Interpretation of Infrared Dynamics in de Sitter
  The infrared dynamics of a light, minimally coupled scalar field in de Sitter
spacetime with Ricci curvature $R=12H$, averaged over horizon sized regions of
physical volume $V_H=\frac{4\pi}{3}\left(\frac{1}{H}\right)^3$, can be
interpreted as Brownian motion in a medium with de Sitter temperature
$T_{DS}=\frac{\hbar H}{2\pi}$. We demonstrate this by directly deriving the
effective action of scalar field fluctuations with wavelengths larger than the
de Sitter curvature radius and generalizing Starobinsky's seminal results on
stochastic inflation. The effective action describes stochastic dynamics and
the fluctuating force drives the field to an equilibrium characterized by a
thermal Gibbs distribution at temperature $T_{DS}$ which corresponds to a de
Sitter invariant state. Hence, approach towards this state can be interpreted
as thermalization. We show that the stochastic kinetic energy of the
coarse-grained description corresponds to the norm of $\partial_\mu\phi$ and
takes a well defined value per horizon volume $\frac{1}{2}\langle
\left(\nabla\phi\right)^2\rangle = - \frac{1}{2}T_{DS}/V_H$. This approach
allows for the non-perturbative computation of the de Sitter invariant stress
energy tensor $\langle T_{\mu\nu} \rangle$ for an arbitrary scalar potential.
