  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 $$\left\{ \begin{aligned} \frac{\partial v}{\partial
t}(x)+{\overline H}\left (\frac{\partial v }{\partial x}(x) \right )=0 &\\ v
(0,x)=f(x)& \end{aligned} \right.$$
