  In this paper we study some thermal properties of quantum field theories in
de Sitter space by means of holographic techniques. We focus on the static
patch of de Sitter and assume that the quantum fields are in the standard
Bunch-Davies vacuum. More specifically, we follow the stochastic motion of a
massive charged particle due to its interaction with Hawking radiation. The
process is described in terms of the theory of Brownian motion in inhomogeneous
media and its associated Langevin dynamics. At late times, we find that the
particle undergoes a regime of slow diffusion and never reaches the horizon, in
stark contrast to the usual random walk behavior at finite temperature.
Nevertheless, the fluctuation-dissipation theorem is found to hold at all
times.
