NONCONVEX HOMOGENIZATION FOR ONE-DIMENSIONAL CONTROLLED RANDOM WALKS IN RANDOM POTENTIAL

• 成果类型: Article
• 署名作者: Yilmaz, Atilla; Zeitouni, Ofer
• 署名单位: Koc University; New York University; Weizmann Institute of Science
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1395
• 发表日期: 2019
• 页码: 36-88
• 关键词: hamilton-jacobi equations; quenched large deviations; Stochastic Homogenization; free-energy

摘要

We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk {X-i} on the set of integers. The cost function is the expectation of the exponential of the path sum of a random stationary and ergodic bounded potential plus theta X-n. The random walk policies are measurable with respect to the random potential, and are adapted, with their drifts uniformly bounded in magnitude by a parameter delta is an element of [0, 1]. Under natural conditions on the potential, we prove that the normalized logarithm of the optimal cost function converges. The proof is constructive in the sense that we identify asymptotically optimal policies given the value of the parameter delta, as well as the law of the potential. It relies on correctors from large deviation theory as opposed to arguments based on subadditivity which do not seem to work except when delta = 0. The Bellman equation associated to this control problem is a second-order Hamilton-Jacobi (HJ) partial difference equation with a separable random Hamiltonian which is nonconvex in theta unless delta = 0. We prove that this equation homogenizes under linear initial data to a first-order HJ equation with a deterministic effective Hamiltonian. When delta = 0, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in theta. In contrast, when delta = 1, the effective Hamiltonian is piecewise linear and nonconvex in theta. Finally, when delta is an element of (0, 1), the effective Hamiltonian is expressed completely in terms of the tilted free energy for the delta = 0 case and its convexity/nonconvexity in theta is characterized by a simple inequality involving delta and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.