Stochastic homogenization of nonconvex Hamilton–Jacobi equations in one space dimension

Abstract

Abstract. We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton-Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.

Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations

“…The following lemma was proved in Armstrong, Tran, Yu [2]. We state it here as it is needed in the proof of the above theorem.…”

“…It is interesting to point out that H s is actually quasi-convex although H is non-convex. This is a specific example of the general "quasi-convexification" phenomenon established in [2]. LetV : [0, 1] → R be a piecewise linear function oscillating between 0 and -1 (see the right graph of Fig.…”

“…Very recently, Armstrong, Tran and Yu [1,2] studied nonconvex stochastic homogenization and derived qualitative properties of H in the general one dimensional case and in some special cases in multi-dimensional spaces. The general case in multi-dimensional spaces is still out of reach.…”

Stochastic Homogenization of Nonconvex Hamilton‐Jacobi Equations: A Counterexample

“…The question of the homogenization of Hamilton-Jacobi equations in the general case where H is not convex in p had remained open until now, and is regularly mentioned in the literature (see, for instance, [2,7,8,14,18,19]). A few particular cases have been treated, for example, the case of level-set convex Hamiltonians (see Armstrong and Souganidis [5]), the case where the law of H is invariant by rotation (this is a direct consequence of Fehrman [12, theorem 1.1]), the one-dimensional case (see Armstrong, Tran, and Yu [8] and Gao [13]), and the case where the law of H satisfies a finite-range condition (see Armstrong and Cardaliaguet [2]).…”

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

“…They also notice the relationship between (i) the convexity of the effective Hamiltonian H(θ) and (ii) the size of the oscillations of V (T x ω) in comparison to the depth of the wells of H(θ) (which is similar to Remark 2.5). Moreover, they give an implicit formula for H(θ) under additional assumptions (see [4,Lemma 5.2]). The proofs in [4] rely on the existence of sublinear correctors in one dimension which is parallel to our approach (see Section 3 for a summary of our proofs), but are otherwise quite different since they use (first-order) nonlinear PDE techniques.…”