Power-law exponent in multiplicative Langevin equation with temporally correlated noise

TL;DR

This paper analytically investigates how temporal correlations in colored noise influence the power-law exponent in continuous-time multiplicative Langevin equations, revealing dependence on noise details unlike in discrete systems.

Contribution

It provides the first analytical insight into the effect of colored noise correlations on power-law exponents in continuous-time multiplicative processes.

Findings

Power-law exponent depends on multiplicative noise details.

Temporal correlation in noise affects power-law behavior.

Contrast with discrete-time systems where autocorrelation time decreases exponent.

Abstract

Power-law distributions are ubiquitous in nature. Random multiplicative processes are a basic model for the generation of power-law distributions. It is known that, for discrete-time systems, the power-law exponent decreases as the autocorrelation time of the multiplier increases. However, for continuous-time ystems, it has not yet been elucidated as to how the temporal correlation affects the power-law behavior. Herein, we have analytically investigated a multiplicative Langevin equation with colored noise. We show that the power-law exponent depends on the details of the multiplicative noise, in contrast to the case of discrete-time systems.