Homogeneization or Slow Diffusion for Random Walks with Random Reflecting Barriers by Smail Alili

the positions 0 = B0 ≤ B1 ≤ B2 ≤ ... ≤ Bn ≤ ... In this last case it moves necessarily up : Bn denotes the position of the nth reflecting barrier. The barriers are assumed to be thrown at random on N, and more precisely the intervals Xn between the barriers are assumed to be i.i.d.. We prove, giving thus a complete answer to a question of Molchanov, that there is homogeneization for xt iff (q/p) X1 has a finite second moment, and that there is slow diffusion iff (q/p)X1 belongs to the domain of attraction of a stable law with index α < 2, i.e. iff it has a regularly varying tail. But (q/p)X1 never has a regularly varying tail. This explains an intriguing result of Solomon concerning partial attraction of x t when the Xi are geometric : partial attraction is actually the best behaviour we can expect in view of