Stochastic equation for a jumping process with long-time correlations

TL;DR

This paper introduces a Markovian, stationary jumping process with power-law autocorrelations, modeling 1/f noise and long-memory effects, and provides an analytical solution for its velocity distribution with long-lasting initial condition influence.

Contribution

It presents a novel stochastic process with long-time correlations and solves the generalized Langevin equation for this process, highlighting its potential to model 1/f noise and long-memory phenomena.

Findings

The process exhibits power-law autocorrelation functions.

The velocity distribution has sharply falling tails.

Memory of initial conditions persists for a long time.

Abstract

A jumping process, defined in terms of jump size distribution and waiting time distribution, is presented. The jumping rate depends on the process value. The process, which is Markovian and stationary, relaxes to an equilibrium and is characterized by the power-law autocorrelation function. Therefore, it can serve as a model of the 1/f noise as well as a model of the stochastic force in the generalized Langevin equation. This equation is solved for the noise correlations 1/t; the resulting velocity distribution has sharply falling tails. The system preserves the memory about the initial condition for a very long time.