Stochastic homogenization for variational solutions of Hamilton-Jacobi equations

Abstract

Let $(\Omega, \mu)$ be a probability space endowed with an ergodic action, $\tau$ of $( {\mathbb R} ^n, +)$. Let $H(x,p; \omega)=H_\omega(x,p)$ be a smooth Hamiltonian on $T^* {\mathbb R} ^n$ parametrized by $\omega\in \Omega$ and such that $ H(a+x,p;\tau_a\omega)=H(x,p;\omega)$. We consider for an initial condition $f\in C^0 ( {\mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$\left\{ \begin{aligned} \frac{\partial u^{ \varepsilon }}{\partial t}(t,x;\omega)+H\left (\frac{x}{ \varepsilon } , \frac{\partial u^\varepsilon }{\partial x}(t,x;\omega);\omega \right )=0 &\\ u^\varepsilon (0,x;\omega)=f(x)& \end{aligned} \right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $\omega \in \Omega$ we have $C^0-\lim u^{\varepsilon}(t,x;\omega)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$\left\{ \begin{aligned} \frac{\partial v}{\partial t}(x)+{\overline H}\left (\frac{\partial v }{\partial x}(x) \right )=0 &\\ v (0,x)=f(x)& \end{aligned} \right.$$