Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium

Abstract: We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.