Lévy walk with parameter dependent velocity: Hermite polynomial approach and numerical simulation

To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Lévy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Lévy walk. Two approaches are compared for some already known results. We also consider the generalized Lévy walk with parameter dependent velocity. Namely, we consider the Lévy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Lévy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement. (paper)