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

## TL;DR

This paper studies the homogenization of a nonconvex Hamilton-Jacobi equation arising from a controlled random walk in a random potential, identifying asymptotically optimal policies and the nature of the effective Hamiltonian.

## Contribution

It provides a constructive proof of homogenization for a nonconvex stochastic PDE and characterizes the effective Hamiltonian's convexity properties across different control regimes.

## Key findings

- Effective Hamiltonian is convex when control is absent (),
- Effective Hamiltonian becomes nonconvex and piecewise linear when control is maximal (
),
- Homogenization results depend on the control parameter
and the potential's properties.

## Abstract

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 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) stochastic 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 deterministic partial differential equation. 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.