  A study of the non-dissipative Brownian motion in vacuum is presented. The
noise source associated to the stochastic process assumed in this work is
vacuum fluctuations of some quantum field capable of interact with a massive
particle. For the associated Fokker-Planck equation we do a perturbation theory
which has terms presenting dynamics similar to that satisfied by the
probability amplitude in Quantum Theory even though the physical interpretation
be classical. In particular classical observables present expected values that
coincide with those calculated using Quantum Mechanics.
