Stochastic nonlinear differential equation generating 1/f noise

TL;DR

This paper derives a novel stochastic nonlinear differential equation that generates 1/f noise across a wide frequency range, providing a new mathematical model for this phenomenon.

Contribution

It introduces the first stochastic differential equation with multiplicative noise that produces 1/f noise, linking point process models to differential equations.

Findings

The derived equation exhibits 1/f noise over a broad frequency range.

Numerical solutions confirm the power-law distribution of the process.

The model demonstrates 1/f noise through constrained diffusion in a finite interval.

Abstract

Starting from the simple point process model of 1/f noise we derive a stochastic nonlinear differential equation for the signal exhibiting 1/f noise in any desirably wide range of frequency. A stochastic differential equation (the general Langevin equation with a multiplicative noise) that gives 1/f noise is derived for the first time. The solution of the equation exhibits the power-law distribution. The process with 1/f noise is demonstrated by the numerical solution of the derived equation with the appropriate restriction of the diffusion of the signal in some finite interval.