The maximum of the local time of a diffusion process in a drifted Brownian potential

We consider a one-dimensional diffusion process $X$ in a $(-\kappa/2)$-drifted Brownian potential for $\kappa\neq 0$. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of $X$ under the annealed law after suitable renormalization when $\kappa \geq 1$. Moreover, we characterize all the upper and lower classes for the hitting times of $X$, in the sense of Paul L\'evy, and provide laws of the iterated logarithm for the diffusion $X$ itself. To this aim, we use annealed technics.