In this paper, we are interested in some questions of Greven and den
Hollander about the rate function I_ηq of quenched large deviations
for random walk in random environment. By studying the hitting times of RWRE,
we prove that in the recurrent case, lim_θ→0+(I_ηq)′′(θ)=+∞, which gives an affirmative answer to a
conjecture of Greven and den Hollander. We also establish a comparison result
between the rate function of quenched large deviations for a diffusion in a
drifted Brownian potential, and the rate function for a drifted Brownian motion
with the same speed