Multiplicative processes and power laws

Abstract

Takayasu, Sato, and Takayasu [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 probability distribution function, 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, Sato, and Takayasu can be extended to exponents $\ensuremath{\mu}>2,$ which explains the ``reason for the lucky coincidence.'' The multidimensional generalization of the results of Takayasu, Sato, and Takayasu and the present status of the problem are briefly summarized. The discovery of stretched exponential tails in the presence of the cutoff introduced by Takayasu, Sato, and Takayasu is explained theoretically. We end by briefly listing applications.