Behaviors of generalized random walks. The model and its evaluations

Behaviors of the generalized random walks (GRW) are directly studied by using computer. The GRW have a non-linear term yielding a non-Markovian character. The non-Markovian character represents memory effects in the processes through the jumping probabilities P̃αN(m|m−α1) between sites m−α1 and m at Nth step. In the computations, the probability that the walker starting at origin arrives site m after N steps, W(m, N), is plotted by varying three parameters. The three parameters specify the jumping probabilities and the memory effects. Furthermore, the P̃αN(m|m−α1)'s are numerically evaluated from the W(m, N). The W(m, N) is compared with the corresponding probability density w(x, t) obtained by taking the continuous limit.