Some Connections Between Stochastic Mechanics, Optimal Control, and Nonlinear Schrödinger Equations

We first recall how the quantum mechanics of N particles is related, in the limit of large N, to certain nonlinear Schrödinger equations, used also to describe the physical effect of Bose–Einstein condensation. We then discuss how, under the influence of Nelson’s stochastic mechanics, a stochastic variational approach to both quantum mechanics and heat diffusion has been developed. We present such topics together with a newer stochastic optimal control approach to Bose–Einstein condensation. Future lines of research in the different areas of mathematics involved in these studies are mentioned.