  The dynamics of particles moving in a medium defined by its relativistically
invariant stochastic properties is investigated. For this aim, the force
exerted on the particles by the medium is defined by a stationary random
variable as a function of the proper time of the particles. The equations of
motion for a single one-dimensional particle are obtained and numerically
solved. A conservation law for the drift momentum of the particle during its
random motion is shown. Moreover, the conservation of the mean value of the
total linear momentum for two particles repelling each other according with the
Coulomb interaction is also following. Therefore, the results indicate the
realization of a kind of stochastic Noether theorem in the system under study.
Possible applications to the stochastic representation of Quantum Mechanics are
advanced.
