Stochastic homogenization for variational solutions of Hamilton–Jacobi equations

Let (Ω, μ) be a probability space endowed with an ergodic action, τ of ( R ⁿ, +). Let H(x,p; ω)=H_ω(x,p) be a smooth Hamiltonian on T^* R ⁿ parametrized by ω∈ Ω and such that H(a+x,p;τₐω)=H(x,p;ω). We consider for an initial condition f∈ C⁰ ( Rⁿ), the family of variational solutions of the stochastic Hamilton-Jacobi equations $$\{ {aligned} {∂ uε}{∂ t}(t,x;ω)+H (x/ ε , ∂ u^ε /∂ x(t,x;ω);ω )=0 & u^ε (0,x;ω)=f(x)& {aligned} .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $ω ∈ Ω$ we have $C^0-lim uε(t,x;ω)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$\{ {aligned} ∂ v/∂ t(x)+{ H} (∂ v /∂ x(x) )=0 & v (0,x)=f(x)& {aligned} .$$