Cost-augmented Schrödinger bridges on graphs are exactly solvable: a Feynman-Kac tilt replaces learned control

We study how to solve cost‑augmented Schrödinger bridges on graphs exactly, offering a solid mathematical alternative to typical reinforcement‑learning and temporal‑difference penalty methods. By inserting state costs directly into the reference Markov process with a Feynman‑Kac tilt, the problem becomes a standard Schrödinger bridge over a modified reference process. This change removes the need for time discretization or iterative policy learning. The computation proceeds by alternating endpoint rescalings and using sparse matrix‑exponential operations that grow only linearly with the size of the network. We test the approach on a protein‑folding model, adding a free‑energy cost to lower the expected energy barrier of transition paths, and on large‑scale network topologies. The

framework gives a robust mathematical foundation for analyzing transition paths in complex chemical systems such as molecular conformation changes, polymer dynamics, and process state transitions. We also provide implementation support and research guidance for simulating high‑dimensional state spaces without the overhead of learned control, guaranteeing exact convergence rates that depend solely on the endpoint coupling.

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