  In the studies of quantum field theory in curved spacetime, the ambiguous
concept of vacuum state and the particle content is a long-standing debatable
aspect. So far it is well known to us that in the background of the curved
spacetime, some privileged class of observers detect particle production in the
suitably chosen vacuum states of the quantum matter fields. In this work we aim
to study the characteristics behaviour of these produced particles in the
background of the de-Sitter (dS) Friedmann-Lama\^{i}tre-Robertson-Walker (FLRW)
Universe (both for $(1+1)$ and $(3+1)$ dimensions) and $(1+1)$-dimensional
Schwarzschild black hole (BH) spacetime, from the point of view of the
respective privileged class of observers. Here the analysis is confined to the
observers who perceive particle excitations in the conformal vacuum. We
consider some test particles in the thermal bath of the produced particles and
calculate the correlation function of the fluctuation of the random force as
exerted by the produced quanta on the test particles. We obtain that the
correlation function abides by the fluctuation-dissipation theorem, which in
turn signifies that the test particles execute Brownian-like motion in the
thermal bath of the produced quanta.
