An Excursion Onto Schrödinger’s Bridges: Stochastic Flows With Spatio-Temporal Marginals

In a gedanken experiment, in 1931/32, Erwin Schrödinger sought to understand how unlikely events can be reconciled with prior laws dictated by the underlying physics. In the process, he posed and solved a celebrated problem that is now named after him – the Schrödinger’s bridge problem (SBP). In this, one seeks to find the “most likely” paths, out of all incompatible paths with the prior, that stochastic particles took while transitioning. The SBP proved to have yet another interpretation, that of the stochastic optimal control problem to steer diffusive particles so as to match specified marginals – soft probabilistic constraints. Interestingly, the SBP is convex and can be solved by an efficient iterative algorithm known as the Fortet-Sinkhorn algorithm. The dual interpretation of the SBP, as an estimation and a control problem, as well as its computational tractability, are at the heart of an ever-expanding range of applications. The purpose of the present work is to expand substantially the type of control and estimation problems that can be addressed following Schrödinger’s dictum, by incorporating termination (killing) of stochastic flows. Specifically, in the context of estimation, we seek the most likely evolution realizing measured spatio-temporal marginals of killed particles. In the context of control, we seek a suitable control action directing the killed process toward spatio-temporal probabilistic constraints. To this end, we derive a new Schrödinger system of coupled, in space and time, partial differential equations to construct the solution of the proposed problem. Further, we show that a Fortet-Sinkhorn type of algorithm is, once again, available to attain the associated bridge. A key feature of our framework is that the obtained bridge retains the Markovian structure in the prior process, and thereby, the corresponding controller takes the form of state feedback.