Nonconvex homogenization for one-dimensional controlled random walks in random potential.

*(English)*Zbl 1415.60121Summary: We consider a finite horizon stochastic optimal control problem for nearest-neighbor random wal \(\{X_i\}\) 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 \(\delta\in[0,1]\). 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 \(\delta\in(0,1)\), 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.

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 \(\delta\in(0,1)\), 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.

##### MSC:

60K37 | Processes in random environments |

93E20 | Optimal stochastic control |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35F20 | Nonlinear first-order PDEs |

##### Keywords:

random walk in random potential; stochastic optimal control; Hamilton-Jacobi; homogenization; corrector; large deviations; tilted free energy
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\textit{A. Yilmaz} and \textit{O. Zeitouni}, Ann. Appl. Probab. 29, No. 1, 36--88 (2019; Zbl 1415.60121)

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