Time：16:30-18:00, Tuesday, May 26 2026
Venue：E14-215
Speaker：Jian-Guo Liu, Duke University
Title：Optimal control formulation of transition path problems for Markov Jump Processes
Abstract：Efficiently characterizing transition paths between metastable states is key to understanding rare events in stochastic systems. We present an optimal control formulation for transition path problems in Markov jump processes over Polish spaces, extending ideas from diffusions to jump dynamics. The framework introduces a controlled transition rate and an unbounded terminal cost at a stopping time to regulate transitions between metastable states. The running cost has an entropic form, and via the Girsanov transform for jump processes, both finite and infinite-horizon problems are unified as an optimal change of measure on the space of càdlàg paths.
The committor function, satisfying an elliptic boundary value problem, yields explicit expressions for the optimal control and the corresponding path measure. Through Γ-convergence, we construct the limiting process with singular transition rates, linked to the Doob-h transform. The resulting optimally controlled process realizes transition paths almost surely, preserving the bridges of the reference process.
