  There is an extensive literature on the dynamic law of large numbers for
systems of quantum particles, that is, on the derivation of an equation
describing the limiting individual behavior of particles inside a large
ensemble of identical interacting particles. The resulting equations are
generally referred to as nonlinear Scr\"odinger equations or Hartree equations,
or Gross-Pitaevski equations. In this paper we extend some of these convergence
results to a stochastic framework. Concretely we work with the Belavkin
stochastic filtering of many particle quantum systems. The resulting limiting
equation is an equation of a new type, which can be seen as a complex-valued
infinite dimensional nonlinear diffusion of McKean-Vlasov type. This result is
the key ingredient for the theory of quantum mean-field games developed by the
author in a previous paper.
