  We investigate Brownian motions of a particle coupled to vacuum fluctuations
of a quantum field. The Unruh effect predicts that an observer in an
accelerated motion sees the Minkowski vacuum as thermally excited. This
addresses the problem of whether a thermal property appears in a perturbative
random motion of a particle in an accelerated motion due to the coupling or
not. We revisit this problem by solving the equation of motion of a particle
coupled to vacuum fluctuations including the radiation reaction force. We
compute a Fourier integral for the variance of the random velocity in a
rigorous manner. Similarly, we consider a particle coupled to vacuum
fluctuations in de Sitter spacetime motivated by the argument that an observer
in de Sitter spacetime sees the Bunch-Davies vacuum as a thermally excited
state with the Gibbons-Hawking temperature. Our investigation clarifies the
condition that the energy equipartition relation arises in the Brownian motions
of a particle.
