Nonconvex Homogenization for 1D Controlled Brownian Motion in a Random Potential

Explore a stochastic optimal control problem for one-dimensional Brownian motion in a random potential through this 55-minute lecture. Delve into the Bellman equation associated with the control problem, examining its viscous Hamilton-Jacobi equation with a random Hamiltonian. Learn about the convergence of the optimal payoff function's growth rate and the homogenization of the Bellman equation. Discover the explicit form of the effective Hamiltonian and its relationship to the tilted quenched free energy of uncontrolled Brownian motion. Investigate the two distinct control regimes marked by the convexity of the effective Hamiltonian. Gain insights into the proof techniques involving large deviations, corrector construction, and identification of asymptotically optimal control policies.