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Suppose {X,} is a random walk in time-random environment with state space Z d , IX, I approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of { X , } is any stability index a. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment. ## Key words Random walks in time-random environments, discrete fractal, Hausdorff dimension, Packing dimension 2000 MR Subject Classification 60J15 1 Preliminaries and Notations There has recently been considerable interest in stochastic processes in random environments, the researches concerning this subject are focused on two cases: one is the Markov chains in time-random environments proceeded by Cogburn [3] and Orey [4], etc. while the other is the random walks in space-random environments put forward by Solomen [5] and Zeitouni [6], [9], etc.. So far all the corresponding results are limited to the limit theory, such as strong law of large numbers, central limit theorem and principle of large deviation. There are seldom results about

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Author

Xiaomin Zhang; Dihe Hu

Publisher

Elsevier Science; Elsevier ; Elsevier B.V.; Elsevier BV; Springer Science and Business Media LLC (ISSN 0252-9602)

Published

2006

Field

Mathematics (Physical Sciences)