Infinite Dimensional Mean-Field Belavkin Equation: Well-posedness and Derivation

We analyze the mean-field limit of a stochastic Schrödinger equation arising in quantum optimal control and mean-field games, where N interacting particles undergo continuous indirect measurement. For the open quantum system described by Belavkin’s filtering equation, we derive a mean-field approximation under minimal assumptions, extending prior results limited to bounded operators and finite-dimensional settings. By establishing global well-posedness via fixed-point methods—avoiding measure-change techniques—we obtain higher regularity solutions. Furthermore, we prove rigorous convergence to the mean-field limit in an infinite-dimensional framework. Our work provides the first derivation of such limits for wave functions in L2(Rd), with implications for simulating and controlling large quantum systems.