Journal of Statistical Physics, Vol. 58, Nos. 3/4, 1990. p. 643-652.
Any ensemble of random walks with symmetric transition probabilities will have symmetric properties. However, any single realization of such a random walk may be asymmetric. In an earlier paper, Weiss and Weissman developed a measure of asymmetry and applied it to random walks in the absence of a field, showing that the degree of asymmetry (in the diffusion limit) is independent of time and that the most probable degree of asymmetry corresponds to the maximum possible. We show in the present paper how the presence of a symmetric
field can change this result, both in making the degree of asymmetry depend on time, and driving the random walk toward a more symmetric state.