Anomalous diffusion in systems driven by the stable Levy noise with a finite noise relaxation time and inertia

TL;DR

This paper investigates how systems influenced by stable Levy noise with inertia and finite noise relaxation time exhibit anomalous diffusion, including subdiffusion, and analyzes the effects of different noise interpretations on distribution tails and escape times.

Contribution

It introduces a comprehensive analysis of Levy-driven systems with inertia and finite relaxation time, extending understanding of distribution convergence and noise interpretation effects.

Findings

Distribution converges to white-noise case over time.

Multiplicative noise causes subdiffusion with finite variance.

Escape time from potential well is computed.

Abstract

Dynamical systems driven by a general L\'evy stable noise are considered. The inertia is included and the noise, represented by a generalised Ornstein-Uhlenbeck process, has a finite relaxation time. A general linear problem (the additive noise) is solved: the resulting distribution converges with time to the distribution for the white-noise, massless case. Moreover, a multiplicative noise is discussed. It can make the distribution steeper and the variance, which is finite, depends sublinearly on time (subdiffusion). For a small mass, a white-noise limit corresponds to the Stratonovich interpretation. On the other hand, the distribution tails agree with the Ito interpretation if the inertia is very large. An escape time from the potential well is calculated.