  We consider the well-travelled problem of homogenization of random integral
functionals. When the integrand has standard growth conditions, the qualitative
theory is well-understood. When it comes to unbounded functionals, that is,
when the domain of the integrand is not the whole space and may depend on the
space-variable, there is no satisfactory theory. In this contribution we
develop a complete qualitative stochastic homogenization theory for nonconvex
unbounded functionals with convex growth. We first prove that if the integrand
is convex and has $p$-growth from below (with $p>d$, the dimension), then it
admits homogenization regardless of growth conditions from above. This result,
that crucially relies on the existence and sublinearity at infinity of
correctors, is also new in the periodic case. In the case of nonconvex
integrands, we prove that a similar homogenization result holds provided the
nonconvex integrand admits a two-sided estimate by a convex integrand (the
domain of which may depend on the space-variable) that itself admits
homogenization. This result is of interest to the rigorous derivation of rubber
elasticity from polymer physics, which involves the stochastic homogenization
of such unbounded functionals.
