We propose a stochastic description of the time dependent quantum Bose-Einstein condensate at zero temperature, within the context of Nelson stochastic mechanics. We describe an infinite particle limit of interacting diffusions which corresponds to the mean field limit in the related quantum system. We are able to extend the framework of Nelson stochastic mechanics to nonlinear systems in particular to the case of the nonlinear Schr\"odinger equation. We also propose how to extend to this nonlinear case the Guerra-Morato variational approach. Our work can also be seen in the context of a mean field limit of McKean–Vlasov processes in a general situation where the drift is a very singular function depending non-trivially on all the particles.