It has been shown that certain types of random walks in random potentials and Brownian motion in Poissonian potentials undergo a phase transition from sub-ballistic to ballistic behavior when the strength of the underlying drift is increased. The ballistic behavior has been manifested by indicating a limiting area for the normalized motion. In the present article, we provide a refined description of this limiting area with a further development for the case of rotation invariant Poissonian potentials.
"A note on the ballistic limit of random motion in a random potential." Electron. Commun. Probab. 13 393 - 400, 2008. https://doi.org/10.1214/ECP.v13-1394