Stochastic homogenization of a class of quasiconvex and possibly degenerate viscous HJ equations in 1d

TL;DR

The paper addresses stochastic homogenization of a possibly degenerate viscous Hamilton-Jacobi equation in one dimension with H(p)=G(p)+\beta V(x,\omega) in a stationary ergodic medium under a scaled hill condition. It develops a deterministic PDE framework to construct correctors with stationary derivatives for $\lambda\ge\beta$ and uses a one-dimensional, ODE-based approach to establish existence/uniqueness and derivative bounds, enabling a definition of the effective Hamiltonian via θ-parameters. It proves homogenization by showing convergence of $u^{\varepsilon}$ to $\bar u$ solving $\partial_t \bar u = {\mathcal{H}}(G)(D\bar u)$, where ${\mathcal{H}}(G)$ is continuous, coercive, and quasiconvex with a flat part on a central interval. The results unify inviscid and viscous cases and extend homogenization to degenerate diffusion with a quasiconvex nonlinearity, under the scaled hill condition.

Abstract

We prove homogenization for possibly degenerate viscous Hamilton-Jacobi equations with a Hamiltonian of the form $G(p)+V(x,ω)$, where $G$ is a quasiconvex, locally Lipschitz function with superlinear growth, the potential $V(x,ω)$ is bounded and Lipschitz continuous, and the diffusion coefficient $a(x,ω)$ is allowed to vanish on some regions or even on the whole $\mathbb{R}$. The class of random media we consider is defined by an explicit scaled hill condition on the pair $(a,V)$ which is fulfilled as long as the environment is not ``rigid''.

Theorems & Definitions (12)

• proof
• proof
• proof
• proof
• proof
• proof : Proof of Proposition \ref{['prop pre-uniqueness']}
• proof
• proof
• proof
• proof