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

Abstract

We give an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with non-convex Hamiltonian.

Strongly mixing smooth planar vector field without asymptotic directions

“…The strongly mixing example we give in this note, allows for analysis in the spirit of [Zil17]. Our example is also related to the homogenization problem of the non-convex HJ equation studied in [FS17,FFZ21].…”

Stochastic homogenization and effective Hamiltonians of HJ equations in one space dimension: The double-well case

“…In particular, when d = 1, homogenization is established in [7,8,18,25] for certain classes of nonconvex Hamiltonians, but the picture is far from being complete. See also [1,6,13] for some (positive and negative) results in higher dimensions.…”

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

“…Our interest in the one-dimensional setting is motivated by two considerations. First, it was shown by counterexamples ([Zil17,FS17,FFZ21]) that, for dimensions d 2, homogenization can fail if the Hamiltonian has a strict saddle point (otherwise being "standard", in particular superlinear as |p| → +∞ uniformly in (x, ω)). Therefore, unlike in the periodic case, there can be no general homogenization result for nonconvex superlinear Hamiltonians in the stationary ergodic setting for d 2, at least, not without additional assumptions on the mixing properties of the environment or the "shape" of the Hamiltonian (e.g., rotational symmetry or homogeneity in the momentum variables).…”