Stochastic homogenization of quasiconvex degenerate viscous HJ equations in 1d
Andrea Davini
TL;DR
This work establishes the stochastic homogenization of a degenerate viscous Hamilton-Jacobi equation in one dimension within a stationary ergodic setting and under a quasiconvex, superlinear Hamiltonian. The authors develop a cell-problem framework, define a critical value $\lambda_0$ through deterministic correctors, and construct random correctors with stationary derivatives to derive the effective Hamiltonian $\mathcal{H}(H)$, which is shown to be locally Lipschitz and quasiconvex. In the strictly quasiconvex regime, they identify a possible flat part on an interval $[\theta^-(\lambda_0),\theta^+(\lambda_0)]$ where $\mathcal{H}(H)(\theta)=\lambda_0$, and provide two independent arguments to bridge different solutions of the cell equation to establish the bottom behavior. The results extend the homogenization theory for viscous HJ equations to degenerate diffusion in 1D random media and establish structural properties of the effective Hamiltonian that are new even in the periodic setting, with broad implications for the qualitative understanding and computation of effective dynamics in heterogeneous media.
Abstract
We prove homogenization for degenerate viscous Hamilton-Jacobi equations in dimension one in stationary ergodic environments with a quasiconvex and superlinear Hamiltonian of fairly general type. We furthermore show that the effective Hamiltonian is quasiconvex. This latter result is new even in the periodic setting, despite homogenization has been known for quite some time.