Multidimensional random walks in random environments with subclassical limiting behavior (Q1077821)

Let \(X_ n\) be the position of a particle at time n in \({\mathbb{R}}^ d\) where the motions of the random walk \(X_ n\) are governed by a random environment. It is shown that it is possible to make the random walk recurrent and such that \(X_ n\sim (\log n)^ 2.\) These results extend previous work for \(d=1\) of F. Solomon, of G. A. Ritter and of Ya. G. Sinaj [see \textit{Ya. G. Sinaj}, Teor. Veroyatn. Primen. 27, 247-258 (1982; Zbl 0497.60065); English translation in Theory Probab. Appl. 27, 247-258 (1982)]. They also make rigorous previous work by \textit{E. Marinari}, \textit{G. Parisi}, \textit{D. Ruelle} and \textit{P. Windey}, Commun. Math. Phys. 89, 1-12 (1983; Zbl 0532.60062). The general results are applied to two special cases, i.e. one dimensional nearest neighbour systems and a family of multidimensional processes constructed from random potentials with Gaussian distributions.