1ST-PASSAGE PROBLEMS IN LINEAR BIASED CORRELATED WALKS

Abstract

The basic difference equations for biased correlated walks on an infinite line are solved by means of the Fourier-Laplace techniques. In terms of these solutions the discrete Laplace transforms of first-passage probabilities with directions are obtained. By using the latter the one-side return probabilities from the positive (negative) side, R+(R-), are obtained as follows: R+ = 1/2(1 + delta) - 1/2-epsilon(1 + 3-delta + epsilon)(1 + epsilon-delta)-1, R = 1/2(1 - delta - epsilon), where delta and epsilon are the degree of correlation and the degree of anisotropy, respectively, with the ranges 0 less-than-or-equal-to delta less-than-or-equal-to 1 and 0 less-than-or-equal-to epsilon less-than-or-equal-to 1-delta. The above results are obtained with the condition that the walker initially arrived at the origin with the right step (positive direction).