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

Abstract

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.

Lévy walk dynamics in non-static media

“…Moreover, the first moment in comoving coordinate x(t) = 0 since R 1 (s) = 0, which indicates the process in comoving coordinate is symmetric. It should be noted that the normalization of the PDF W (x, t) can also be verified through Hermite polynomials expansion and the result of R 0 = 1/ √ π, the detailed calculations of which can be found in [40,41].…”

“…Intuitively, a superdiffusive type of Lévy walk in physical coordinate can keep the diffusion exponent, which is a kind of stable property mentioned in [55]. Further it can be found in [40] that the MSD of ordinary Lévy walk can be obtained through multiplying y 2 (t) in (42) with 2/(2 − α) for α ∈ (0, 1) ∪ (1, 2). Therefore one can conclude that the contracting exponential medium cannot localize superdiffusive or ballistic types of Lévy walks, and it can only make the diffusion constant become slower comparing to the ordinary case instead of changing the diffusion exponent.…”

“…In the following, to theoretically obtain the behaviors of moments in comoving and physical coordinates, we utilize Hermite orthogonal polynomials to approach PDF [40][41][42].…”

“…We first build the model and derive the governing equations. Then using Hermite polynomials expansion established in [40] solves the equation to calculate the statistical observables. The method of Hermite orthogonal polynomials expansion is good at calculating MSD even for some cases that are unsolvable by ordinary method of integral transform, such as Lévy walks moving under external potential [41,42].…”

Gaussian Process and Levy Walk under Stochastic Non-instantaneous Resetting and Stochastic Rest

“…random variables known as waiting time and jump length are independent with each other [16], while the dynamic of Lévy walk is space-time coupled through finite propagation speed [15]. Recently, the methods of Hermite polynomial expansions are established to solve some issues of Lévy walk [17], essentially being effective for Lévy walk with external potential [18,19]. Both CTRW and Lévy walk have many applications in finance [20], ecology [21], and biology [22].…”