Stochastic Control and Optimal Mass Transport- A Schroedinger Bridge Approach
Abstract
We propose to study key issues at the interface between stochastic control and optimal mass transport. The setting is that of the Schrodinger Bridge Problem (SBP), which seeks laws on the space of sample paths of stochastic processes, with marginal constraints at various points in time. Besides growing interest in the topic due to applications in control, image-signal processing, machine learning, and medicine, when developed in the context of discrete spaces (networks), it provides new insights and tools to address the controllability and robustness of a given network. Recent research has pointed to the deep connections among large deviations theory (rare events), fluctuation theorems (nonequilibrium thermodynamics), and Nelson’s stochastic mechanics. It turns out that Schrodinger Bridges (SB) s are closely connected to optimal mass transport (OMT), which in turn has been used as a tool to quantify trends to equilibrium in dissipative partial differential equations. Furthermore, OMT has a rich geometric structure which may be exploited for understanding constrained stochastic flows. In particular, OMT leads to a Riemannian-type metric on the space of probability distributions from which various types of partial differential equations such as the Fokker-Planck equation, may be studied as gradient flows. In fact, in the SBP-OMT literature, concavity properties of the entropy functional along stochastic flows have been proposed as a proxy for curvature. Our motivation, and what sets our study apart, is that we propose these notions (OMT-SBP geometry, Bakry-Emery theory, concavity of entropy and Wasserstein distance) to study and optimize the controllability and robustness of systems (both continuous and discrete), more specifically, the ability of a networked system to withstand perturbations and thereafter return to a stationary state.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 29, 2024
- Source ID
- FA95502310096
Entities
People
- Allen Tannenbaum
Organizations
- Air Force Office of Scientific Research
- Research Foundation for the State University of New York
- United States Air Force