Fluctuations in multiplicative systems with jumps

TL;DR

This paper investigates the fluctuation behavior of multiplicative systems driven by Levy stable noise, analyzing covariance decay, distribution tails, and diffusion properties in systems with jumps and power-law noise.

Contribution

It introduces a detailed analysis of covariance and distribution behaviors in multiplicative systems with Levy stable noise, including truncated and stable cases, and explores their diffusion characteristics.

Findings

Covariance decays exponentially with truncated Levy noise.

Stable Levy noise leads to weakly stretched exponential covariance.

Systems exhibit power-law tails with Gaussian-like central parts.

Abstract

Fluctuation properties of the Langevin equation including a multiplicative, power-law noise and a quadratic potential are discussed. The noise has the Levy stable distribution. If this distribution is truncated, the covariance can be derived in the limit of large time; it falls exponentially. Covariance in the stable case, studied for the Cauchy distribution, exhibits a weakly stretched exponential shape and can be approximated by the simple exponential. The dependence of that function on system parameters is determined. Then we consider a dynamics which involves the above process and obey the generalised Langevin equation, the same as for Gaussian case. The resulting distributions possess power-law tails - that fall similarly to those for the driving noise - whereas central parts can assume the Gaussian shape. Moreover, a process with the covariance 1/t at large time is constructed and the corresponding dynamical equation solved. Diffusion properties of systems for both covariances are discussed.