Anomalous diffusion resulting from strongly asymmetric random walks

We present a model of one-dimensional asymmetric random walks. Random walkers alternate between flights ~steps of constant velocity! and sticking ~pauses!. The sticking time probability distribution function ~PDF! decays as P(t);t. Previous work considered the case of a flight PDF decaying as P(t);t @Weeks et al., Physica D 97, 291 ~1996!#; leftward and rightward flights occurred with differing probabilities and velocities. In addition to these asymmetries, the present strongly asymmetric model uses distinct flight PDFs for leftward and rightward flights: PL(t);t 2m and PR(t);t , with mÞh . We calculate the dependence of the variance exponent g (s;t) on the PDF exponents n , m , and h . We find that g is determined by the two smaller of the three PDF exponents, and in some cases by only the smallest. A PDF with decay exponent less than 3 has a divergent second moment, and thus is a Lévy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive (1,g,2). When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive (g,1); this is in contrast with the case with m5h. @S1063-651X~98!01205-7#