Stochastic homogenization of viscous superquadratic Hamilton\u2013Jacobi equations in dynamic random environment

Wenjia Jing,Panagiotis E Souganidis,Hung V Tran

doi:10.1186/s40687-016-0090-9

Wenjia Jing, Panagiotis E Souganidis + Show 1 more

Open Access

https://doi.org/10.1186/s40687-016-0090-9

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Abstract

We study the qualitative homogenization of second-order Hamilton–Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion.

Highlights

We study the homogenized behavior of the solution uε = uε(x, t, ω) to the second-order Hamilton–Jacobi equation ⎧ ⎪⎨ uεt − εtr A x ε, t ε, ω D2uε +HDuε, x, t, ω εε =0⎪⎩ uε = u0 in Rn × (0, +∞), (1)
1 Background We study the homogenized behavior of the solution uε = uε(x, t, ω) to the second-order Hamilton–Jacobi equation
We extend and combine the methodologies of [22] and [5,6,7] and obtain stochastic homogenization for general viscous Hamilton–Jacobi equations in dynamic random environments

Summary

Introduction

[14] yields non-convex Hamiltonians with space-time random potentials for which the Hamilton–Jacobi equation does not homogenize. This is a subadditive, stationary function which, in view of the subadditive ergodic theorem, has a homogenized limit, that identifies the convex conjugate of the effective Hamiltonian H.

Results

Conclusion