An excursion onto Schrödinger's bridges: Stochastic flows with spatio-temporal marginals

Abstract

The purpose of the present work is to expand substantially the type of control and estimation problems that can be addressed following the paradigm of Schrödinger bridges, 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 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.

Figures (2)

• Figure 1: Illustration of diffusion with losses: The top sub-figure shows two sample paths of particles initially in the primary space ${\mathcal{X}}$ (a path for a surviving particle and another for a killed one) with a drawing of the initial marginal $\rho_0$ and also of the final marginal of surviving particles. The bottom sub-figure shows the second segment of the killed path and the spatio-temporal marginal $Q_t$ of killed particles in the coffin space $\widetilde{{\mathcal{X}}}$.
• Figure 2: Top: One-time marginals of ${\mathbf P}^\star$ restricted to the primary space. Bottom: One-time marginals of $Q_t$ (black), the spatio-temporal marginal of the particles in the coffin space as obtained from the Fortet-Sinkhorn iteration \ref{['eq:sink1']} (shaded surface).

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof