Linear stochastic dynamics with nonlinear fractal properties

TL;DR

This paper reviews stochastic processes with multiplicative noise that produce power-law distributed intermittency, highlighting their applications across various fields and analyzing burst duration statistics.

Contribution

It provides a comprehensive review of the physical mechanisms and statistical properties of intermittent bursts in multiplicative noise processes across multiple disciplines.

Findings

Power-law probability density distributions characterize intermittency.

The distribution and duration of bursts are quantitatively analyzed.

Applications include population dynamics, finance, and internet traffic.

Abstract

Stochastic processes with multiplicative noise have been studied independently 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 present a review of applications on population dynamics, epidemics, finance and insurance applications with relation to ARCH(1) process, immigration and investment portfolios and the internet. We highlight the common physical mechanism and summarize the main known results. The distribution and statistical properties of the duration of intermittent bursts are also characterized in details.