I will present a stochastic optimal control problem for Brownian motion $(X_s)_{0 \le s \le t}$ on the real line. The payoff function is the exponential expectation of the path integral of a random potential (of amplitude $\beta$) plus $\theta X_t$. The control policies are adapted drifts that are uniformly bounded in magnitude by another parameter $c$. The Bellman equation associated with this control problem is a viscous HJ equation with a random Hamiltonian which is nonconvex in $\theta$. Under natural conditions on the potential, I will show that the rate of growth of the optimal payoff function converges (as $t \to \infty$) almost surely to some deterministic quantity $\overline H_{\beta,c}(\theta)$. It follows that the associated Bellman equation homogenizes to an inviscid HJ equation and $\overline H_{\beta,c}$ is precisely the effective Hamiltonian. Moreover, this quantity is explicit in terms of the tilted quenched free energy of (uncontrolled) Brownian motion in a random potential and it is convex in $\theta$ iff $\beta \ge \frac{1}{2}c^2$, thereby marking two qualitatively distinct control regimes. The proofs involve large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal control policies.
Based on joint work with E. Kosygina and O. Zeitouni (Comm. PDE 2020).
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