Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media

Abstract

This paper considers the problem of homogenization of a class of convex Hamilton-Jacobi equations in spatio-temporal stationary ergodic environments. Special attention is placed on the interplay between the use of the Subadditive Ergodic Theorem and continuity estimates for the solutions that are independent of the oscillations in the equation. Moreover, an inf-sup formula for the effective Hamiltonian is provided.

Stochastic Homogenization of Nonconvex Hamilton‐Jacobi Equations: A Counterexample

“…They have proved independently that when H is convex with respect to p, then u ǫ converges P-almost surely to the unique solution of a system of the form ∂ t u(x, t) +H(Du(x, t)) = 0 in R n × (0, +∞) u(x, 0) = 0 in R n whereH is the effective Hamiltonian. This result has been extended to various frameworks, still under the assumption that the Hamiltonian is convex in p (see [19,15,18,17,22,4,6]). Quantitative results about the speed of convergence have been obtained in [3,20,2].…”

De Giorgi Techniques Applied to the Hölder Regularity of Solutions to Hamilton–Jacobi Equations

“…The first Hölder regularity result of this kind was obtained by Schwab, in [10], in the case of a convex Hamiltonian. The result was a key ingredient to perform the stochastic homogenization of Hamilton-Jacobi equations in Stationary Ergodic Spatio-Temporal media.…”

“…So, it follows from (8), (10), and (11) that the following relation holds for each k ≥ 1, where C(N, p, Λ) > 0 is some absolute constant which depends only on N , p and Λ.…”

Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment

“…For firstorder equations with superlinear Hamiltonians, homogenization results were proved by Schwab [26]. Recently, the authors [17] established homogenization for linearly growing Hamiltonians that are periodic in space and stationary ergodic in time.…”