Abstract:

Suppose {X_n} is a random walk in time-random environment with state space Z^d, |X_n| approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {X_n} is any stability index α. 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