Local and occupation time of a particle diffusing in a random medium

We consider a particle moving in a one-dimensional potential which has a symmetric deterministic part and a quenched random part. We study analytically the probability distributions of the local time (spent by the particle around its mean value) and the occupation time (spent above its mean value) within an observation time window of size t. In the absence of quenched randomness, these distributions have three typical asymptotic behaviors depending on whether the deterministic potential is unstable, stable, or flat. These asymptotic behaviors are shown to get drastically modified when the random part of the potential is switched on, leading to the loss of self-averaging and wide sample to sample fluctuations.