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

Abstract

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.

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.…”

“…This began with the work of Souganidis [21] and of Rezakhanlou and Tarver [20] who independently proved homogenization in the case of H convex in general stationary ergodic random media. The literature is vast, see for example [19], [9], and [16] for more thorough reviews of the history of the subject (especially with focus on the viscous case).…”

A Parallel Algorithm for the Constrained Shortest Path Problem on Lattice Graphs

“…The constrained first passage percolation problem is a discrete analog to Hamilton-Jacobi equation. The large time behaviors of its solutions are extensively studied and homogenization results are obtained for a class of Hamiltonians [2,13,14,26]. Fluctuations in dimension one are of order t [24] while in higher dimensions they are of lower order although only the logarithmic improvement to the bound has been achieved so far [19].…”

Stochastic homogenization of certain nonconvex Hamilton–Jacobi equations

“…The first result was contributed by Davini and Kosygina [11] when the Hamiltonian is piecewise level-set convex and pinned at junctions. With the help of a probabilistic approach, Kosygina, Yilmaz and Zeitouni [24] (see also Yilmaz and Zeitouni [35]) established the homogenization for a Hamiltonian that takes a 'W' shape and the potential function satisfies certain valley-hill assumption.…”