Abstract: In this talk, I will present a stochastic optimal control formulation for (i) transition path problems in an infinite time horizon, specifically for Markov jump processes on Polish spaces, and (ii) mean field games in a finite time horizon. Transition paths connecting metastable states in a stochastic model are rare events that appear in many applications in science and engineering. An unbounded terminal cost at a stopping time, along with a controlled transition rate, regulates the transitions between metastable states. To maintain the original bridges after control, the running cost adopts an entropic form for the control velocity. However, the unbounded terminal cost leads to a singular optimal control and presents difficulties in the Girsanov transform. Gamma-convergence techniques and passing the limit in the corresponding Martingale problem allow us to obtain a singular optimally controlled transition rate. We demonstrate that the committor function, which solves a backward equation with specific boundary conditions, provides an explicit formula for the optimal path measure. The optimally controlled process realizes the transition paths almost surely but without altering the bridges of the original process. This stochastic optimal control formulation is also applicable to mean field games.