  Fix strictly increasing right continuous functions with left limits $W_i:\bb
R \to \bb R$, $i=1,...,d$, and let $W(x) = \sum_{i=1}^d W_i(x_i)$ for $x\in\bb
R^d$. We construct the $W$-Sobolev spaces, which consist of functions $f$
having weak generalized gradients $\nabla_W f = (\partial_{W_1}
f,...,\partial_{W_d} f)$. Several properties, that are analogous to classical
results on Sobolev spaces, are obtained. $W$-generalized elliptic and parabolic
equations are also established, along with results on existence and uniqueness
of weak solutions of such equations. Homogenization results of suitable random
operators are investigated. Finally, as an application of all the theory
developed, we prove a hydrodynamic limit for gradient processes with
conductances (induced by $W$) in random environments.
