Spurious memory in non-equilibrium stochastic models of imitative behavior

TL;DR

This paper investigates the origin of long-range memory in non-equilibrium stochastic models, distinguishing between genuine and spurious memory, and proposes empirical tests based on burst duration PDFs to identify the source of memory.

Contribution

It compares stochastic differential equation models with fractional Brownian motion, identifying a power-law exponent as a marker for spurious memory in non-equilibrium systems.

Findings

Power-law exponent 3/2 in burst duration PDFs indicates spurious memory.

Differences between SDE-driven processes and fBm can be empirically tested.

Method can help identify the origin of long-range memory in complex systems.

Abstract

The origin of the long-range memory in the non-equilibrium systems is still an open problem as the phenomenon can be reproduced using models based on Markov processes. In these cases a notion of spurious memory is introduced. A good example of Markov processes with spurious memory is stochastic process driven by a non-linear stochastic differential equation (SDE). This example is at odds with models built using fractional Brownian motion (fBm). We analyze differences between these two cases seeking to establish possible empirical tests of the origin of the observed long-range memory. We investigate probability density functions (PDFs) of burst and inter-burst duration in numerically obtained time series and compare with the results of fBm. Our analysis confirms that the characteristic feature of the processes described by a one-dimensional SDE is the power-law exponent $3/2$ of the burst or inter-burst duration PDF. This property of stochastic processes might be used to detect spurious memory in various non-equilibrium systems, where observed macroscopic behavior can be derived from the imitative interactions of agents.