  A thermal interpretation of the stochastic formalism of a slow-rolling scalar
field in de Sitter (dS) is given. We construct a correspondence between Hubble
patches of dS and particles living in another space called an abstract space.
By assuming a dual description of scalar fields and classical mechanics in the
abstract space, we show that the stochastic evolution of the infrared part of
the field is equivalent to the Brownian motion in the abstract space filled
with a heat bath of massless particles. The 1st slow-roll condition and the
Hubble expansion are also reinterpreted in the abstract space as the speed of
light and a transfer of conserved energy, respectively. Inspired by this, we
sketch quantum emergent particles, which may realize the Hubble expansion by an
exponential particle production. This gives another meaning of dS entropy as
entropy per Hubble volume.
