  We prove stochastic homogenization for integral functionals defined on
Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate
growth condition of the form \begin{equation*}
  c|\xi A(\omega,x)|^p\leq f(\omega,x,\xi)\leq |\xi
A(\omega,x)|^p+\Lambda(\omega,x) \end{equation*} for some $p\in (1,+\infty)$
and with a stationary and ergodic diagonal matrix $A$ such that its norm and
the norm of its inverse satisfy minimal integrability assumptions. We also
consider the convergence when Dirichlet boundary conditions or an obstacle
condition are imposed. Assuming the strict convexity and differentiability of
$f$ with respect to its last variable, we further prove that the homogenized
integrand is also strictly convex and differentiable. These properties allow us
to show homogenization of the associated Euler-Lagrange equations.
