Schroedinger bridges: steering of stochastic systems classical and quantum
Abstract:
The classical Schroedinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior, and the law dictates a controlled path that abides by the specified marginals. Schreodinger proved that the optimal steering of the density between the two end points is effected by a multiplicative functional transformation of the prior; this transformation represents an automorphism on the space of probability measures and has since been studied by Fortet, Beurling and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. Ultimately, in all of the above Schrodinger’s question relates to a corresponding stochastic control problem to steer a stochastic system from an initial to a final distribution. In the talk we will begin with a treatment of the Schroedinger bridge problem for Markov chains, Quantum channels, and then for classical stochastic systems where we will present in the linear-quadratic case (linear dynamics, Gaussian evolution) the solution to the minimum energy steering problem in closed form. We will discuss potential applications in active suppression of noise and in steering swarms of inertial diffusive particles. The presentation will be based on joint work with Michele Pavon and with Yongxin Chen.