2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74210In this paper, we are interested in some questions of Greven and den Hollander about the rate function $I\_η^q$ of quenched large deviations for random walk in random environment. By studying the hitting times of RWRE, we prove that in the recurrent case, $\lim\_{θ\to 0^+}(I\_η^q)''(θ)=+\infty$, 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.ProbabilityAMS (2000) Classification: 60K37, 60F10, 60J60text