Directed random walks in continuous space

The investigation on diffusion with directed motion in a two-dimensional continuous space is completed by using the model of the continuous directed random walks. The average square end-to-end distance <R2(t)> approximately t(2nu) is calculated. The results show that this type of walks belongs asymptotically to the same class (nu=1.0) as the ballistic motions. For short time, we observe a crossover from purely random walks (nu=0.5) to ballistic motions (nu=1.0). The dependence of the crossover on the direction parameter theta is studied. There exists a scaling relation of the form <R2(t)> approximately tf(t/theta(-2)). The return probability P00(t) is also investigated and the scaling form similar to <R2(t)> is obtained.