Stochastic Description of a Bose-Einstein Condensate

In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose-Einstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross-Pitaevskii scaling, which allows to give a theoretical proof of Bose-Einstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of the single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random "interaction-set" with probability one. Moreover we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate.