Stochastic homogenization of nondegenerate viscous HJ equations in 1d
Andrea Davini
TL;DR
This work proves stochastic homogenization for a nondegenerate viscous Hamilton–Jacobi equation in one dimension with a stationary ergodic, generally nonconvex, superlinear Hamiltonian. The authors develop a PDE-centered framework built around stationary correctors for the cell problem $a(x,\omega)u''+H(x,u',\omega)=\lambda$, and they introduce a dual ODE perspective linking correctors to stationary derivatives. A novel bridging argument eliminates gaps between stationary solutions, yielding a full characterization of the effective Hamiltonian ${\mathcal{H}}(H)$ and ensuring the homogenization limit exists and is governed by a continuous, coercive, locally Lipschitz, and superlinear function. The results extend the homogenization theory beyond convexity and hill/valley conditions, capturing general nondegenerate diffusion in 1D random media and providing a robust foundation for further studies in stochastic HJ equations with nonconvex Hamiltonians. The key contributions include the existence and structure of correctors, the bridging technique to control flat parts of the effective Hamiltonian, and a complete homogenization theorem with precise regularity properties of ${\mathcal{H}}(H)$.
Abstract
We prove homogenization for a nondegenerate viscous Hamilton-Jacobi equation in dimension one in stationary ergodic environments with a superlinear (nonconvex) Hamiltonian of fairly general type. The version of the paper herein posted is identical to the one submitted to the journal *** for publication on the 26th of February, 2024. One of the crucial idea for the proof corresponds to Theorem 4.2. This theorem, together with its proof, already appeared, in essential identical form, in the ArXiv e-print 2306.12145, version 1 (posted on the 21st of June 2023), see Theorem 4.5 therein.