Infrared dynamics of a light scalar field in de Sitter

Inertial observers in de Sitter are surrounded by a horizon and see thermal fluctuations. To them, a massless scalar field appears to follow a random motion but any attractive potential, no matter how weak, will eventually stabilize the field. We study this thermalization process in the static patch (the spacetime region accessible to an individual observer) via a truncation to the low frequency spectrum. We focus on the distribution of the field averaged over a subhorizon region. At timescales much longer than the inverse temperature and to leading order in the coupling, we find the evolution to be Markovian, governed by the same Fokker-Planck equation that arises when the theory is studied in the inflationary setup.