  We study a symmetric diffusion process on $\mathbb{R}^d$, $d\geq 2$, in
divergence form in a stationary and ergodic random environment. The
coefficients are assumed to be degenerate and unbounded but satisfy a moment
condition. We derive upper off-diagonal estimates on the heat kernel of this
process for general speed measure. Lower off-diagonal estimates are also shown
for a natural choice of speed measure, under an additional mixing assumption on
the environment. Using these estimates, a scaling limit for the Green's
function is proven.
