  The Brownian motion of a point particle induced by quantum vacuum
fluctuations of a massless real scalar field in Einstein's universe is studied.
By assuming the small displacement condition, the dispersion in the momentum
and position of a point particle coupled to the massless scalar field are
obtained. As a consequence of the homogeneity and isotropy properties of the
Einstein's Universe, we find that all components of these physical observables
are identical. We also examine divergent behaviors associated to the physical
momentum and position dispersions, which we attribute to the $R^1\times S^3$
compact topology of the spacetime. Finally, based on the small displacement
condition assumed, we analyze the limit of validity of our investigation.
