  In this paper, we are interested in some questions of Greven and den
Hollander about the rate function $I\_{\eta}^q$ of quenched large deviations
for random walk in random environment. By studying the hitting times of RWRE,
we prove that in the recurrent case, $\lim\_{\theta\to
0^+}(I\_{\eta}^q)''(\theta)=+\infty$, 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.
