Abstract
We study the one-dimensional random walk of a particle in the presence of a short-range correlated quenched random field of jump lengths l(x) drawn from a Lévy type distribution 
with 0<f<2. We find the stochastic dynamics to be characterized by a novel length-time scaling relation that is caused by an effective jump-length distribution 
in the stationary state, which decays more rapidly than p(l), i.e. 
. For 
, g becomes larger than 2 and the particle diffuses normally although p(l) has no finite second moment. A scaling theory is developed that describes the dynamical crossover from the annealed to the quenched situation.