Multiplicative processes and power laws

TL;DR

This paper revisits stochastic multiplicative processes, explaining how they produce power law distributions and extending previous results to broader parameter regimes, with implications for various physical and complex systems.

Contribution

It extends Takayasu et al.'s formulation to exponents greater than 2 and summarizes multidimensional generalizations and applications of multiplicative processes.

Findings

Power law probability density functions arise in multiplicative noise processes.

The characteristic function approach can be extended to exponents greater than 2.

Stretched exponential tails occur with introduced cut-offs.

Abstract

[Takayasu et al., Phys. Rev.Lett. 79, 966 (1997)] revisited the question of stochastic processes with multiplicative noise, which have been studied in several different contexts over the past decades. We focus on the regime, found for a generic set of control parameters, in which stochastic processes with multiplicative noise produce intermittency of a special kind, characterized by a power law probability density distribution. We briefly explain the physical mechanism leading to a power law pdf and provide a list of references for these results dating back from a quarter of century. We explain how the formulation in terms of the characteristic function developed by Takayasu et al. can be extended to exponents $\mu >2$, which explains the ``reason of the lucky coincidence''. The multidimensional generalization of (\ref{eq1}) and the available results are briefly summarized. The discovery of stretched exponential tails in the presence of the cut-off introduced in \cite{Taka} is explained theoretically. We end by briefly listing applications.