Anomalous diffusion for a correlated process with long jumps

TL;DR

This paper investigates the diffusion behavior of a correlated jump process with long-tail velocity distributions, revealing conditions under which the position variance remains finite yet exhibits anomalously fast growth.

Contribution

It introduces a modified jumping process with stable tails and finite correlations, demonstrating how these features influence anomalous diffusion and variance behavior.

Findings

Stationary solutions decay over time but can be stabilized with process modifications.

Position variance can be finite despite superdiffusive growth.

Superimposed Ornstein-Uhlenbeck-Lévy processes lead to divergent variances.

Abstract

We discuss diffusion properties of a dynamical system, which is characterised by long-tail distributions and finite correlations. The particle velocity has the stable L\'evy distribution; it is assumed as a jumping process (the kangaroo process) with a variable jumping rate. Both the exponential and the algebraic form of the covariance -- defined for the truncated distribution -- are considered. It is demonstrated by numerical calculations that the stationary solution of the master equation for the case of power-law correlations decays with time, but a simple modification of the process makes the tails stable. The main result of the paper is a finding that -- in contrast to the velocity fluctuations -- the position variance may be finite. It rises with time faster than linearly: the diffusion is anomalously enhanced. On the other hand, a process which follows from a superposition of the Ornstein-Uhlenbeck-L\'evy processes always leads to position distributions with a divergent variance which means accelerated diffusion.