Paradoxical non-Gaussian behavior in fractional Laplace motion with drift

TL;DR

This paper investigates the complex behavior of fractional Laplace motion with drift, revealing how internal and external drifts differently influence diffusion and the probability density function, with implications for modeling non-Gaussian stochastic processes.

Contribution

It provides a detailed analysis of the statistical properties of fractional Laplace motion with drift, highlighting the distinct effects of internal and external drifts on diffusion and PDF characteristics.

Findings

External drift does not affect MSD.

Internal drift causes normal diffusion at long times.

PDF exhibits a Gaussian core with non-Gaussian tails.

Abstract

We study fractional Laplace motion (FLM) obtained from subordination of fractional Brownian motion to a gamma process, in the presence of an external drift that acts on the composite process or of an internal drift acting solely on the parental process. We derive the statistical properties of this FLM process and find that the external drift does not influence the mean-squared displacement (MSD), whereas the internal drift leads to normal diffusion, dominating at long times in the subdiffusive Hurst exponent regime. We also investigate the intricate properties of the probability density function (PDF), demonstrating that it possesses a central Gaussian region, whose expansion in time is influenced by FBM's Hurst exponent. Outside of this region the PDF follows a non-Gaussian pattern. The kurtosis of this FLM process converges toward the Gaussian limit at long times insensitive to the extreme non-Gaussian tails. Additionally, in the presence of the external drift, the PDF remains symmetric and centered at $x=vt$. In contrast, for the internal drift this symmetry is broken. The results of our computer simulations are fully consistent with the theoretical predictions. The FLM model is suitable for describing stochastic processes with a non-Gaussian PDF and long-ranged correlations of the motion.