Mathematical theory of quantum stochastic filtering

The theory of stochastic quantum filtering (or quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author only recently. This new theory led to numerous further developments including the rigorous derivation of filtering equations from the basic principles, the corresponding law of large number limits (propagation of chaos), fractional quantum mechanics of open systems, quantum dynamic and mean-field games. In the talk, we present some basic results and ideas from this story. The presentation will be mostly based on the survey paper Vassili N. Kolokoltsov “Quantum filtering and propagation of chaos for open quantum systems, with applications to quantum feedback control and quantum mean-field games” http://arxiv.org/abs/2607.08507, to appear in UMN.