Asymptotic power law of moments in a random multiplicative process with weak additive noise

TL;DR

This paper investigates the universal power-law behavior of moments in a random multiplicative process with weak additive noise, using a Langevin-type model to clarify the underlying mechanism and its implications for complex systems.

Contribution

It provides a detailed analysis of the power-law scaling of moments in such processes, demonstrating the universality of the mechanism through both approximate and exact treatments.

Findings

Identifies the power-law scaling of moments as a universal feature.

Clarifies the mechanism behind the power-law behavior.

Links the findings to phenomena like noisy intermittency and spatio-temporal chaos.

Abstract

It is well known that a random multiplicative process with weak additive noise generates a power-law probability distribution. It has recently been recognized that this process exhibits another type of power law: the moment of the stochastic variable scales as a function of the additive noise strength. We clarify the mechanism for this power-law behavior of moments by treating a simple Langevin-type model both approximately and exactly, and argue this mechanism is universal. We also discuss the relevance of our findings to noisy on-off intermittency and to singular spatio-temporal chaos recently observed in systems of non-locally coupled elements.