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

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 θXn. 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 δ ∈ [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 δ, 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 δ = 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 θ unless δ = 0. We prove that this equation homogenizes under linear initial data to a first-order HJ deterministic partial differential equation. When δ = 0, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in θ. In contrast, when δ = 1, the effective Hamiltonian is piecewise linear and nonconvex in θ. Finally, when δ ∈ (0, 1), the effective Hamiltonian is expressed completely in terms of the tilted free energy for the δ = 0 case and its convexity/nonconvexity in θ is characterized by a simple inequality involving δ and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.

An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity

“…The question of the convergence for the viscous Hamilton-Jacobi equation without convexity has remained open. Several recent works have given examples of non-convex Hamiltonians where homogenization holds [1,6,16,22]. In this note we extend the work of the third author [23] and show that homogenization may fail for (1), with all the standard assumptions of the literature except convexity.…”

“…Davini and Kosygina [7] considered Hamiltonians with one or more "pinning points" p where H is constant almost surely, and convex in between. Yılmaz and Zeitouni [22] (discrete case) and Kosygina, Yılmaz, and Zeitouni [16] (continuous case) have proven homogenization for a special nonconvex Hamiltonian H(p, x) = 1 2 |p| 2 − c|p| + V (x, ω). A fundamental limitation to these efforts was discovered by the third author [23], who found an example of a first order non-convex Hamilton-Jacobi equation for which homogenization fails.…”

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

“…The above approach was first proposed and implemented in [YZ17] in the discrete setting where the BM in the control problem (1.7) is replaced by a random walk, the analog of Theorem 1.2 is proved for a viscous Hamilton-Jacobi partial difference equation and the effective Hamiltonian is shown to have the same structure as in this paper. However, as it is often the case, the arguments in the continuous formulation differ noticeably.…”

“…However, as it is often the case, the arguments in the continuous formulation differ noticeably. We believe that some of the ideas in [YZ17] and this paper can be extended to more general settings, for example, to Hamiltonians of the form (1.10) in one or more dimensions.…”

Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension

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We prove homogenization for a class of nonconvex viscous Hamilton-Jacobi equations in stationary ergodic random environment in one space dimension. The result concerns Hamiltonians of the form Gppq `V px, ωq, where the nonlinearity G is a minimum of two or more convex functions with the same absolute minimum, and the potential V is a stationary process satisfying an additional "valley and hill" condition introduced earlier by A. Yilmaz and O. Zeitouni [27]. Our approach is based on PDE methods and does not rely on representation formulas for solutions.

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