Spectral content of fractional Brownian motion with stochastic reset

TL;DR

This paper derives an exact expression for the power spectral density of fractional Brownian motion with stochastic resetting, revealing distinct asymptotic behaviors for sub- and super-diffusive regimes.

Contribution

It provides the first exact formula for the PSD of reset fBm and uncovers the universal high-frequency tail exponent for superdiffusive cases.

Findings

PSD exhibits a power-law tail with exponent 2H+1 for H<1/2

PSD tail exponent is 2 for H≥1/2, independent of H

Large frequency behavior differs markedly between sub- and super-diffusive fBm

Abstract

We analyse the power spectral density (PSD) $S_T(f)$ (with $T$ being the observation time and $f$ is the frequency) of a fractional Brownian motion (fBm), with an arbitrary Hurst index $H \in (0,1)$, undergoing a stochastic resetting to the origin at a constant rate $r$ - the resetting process introduced some time ago as an example of an efficient, optimisable search algorithm. To this end, we first derive an exact expression for the covariance function of an arbitrary (not necessarily a fBm) process with a reset, expressing it through the covariance function of the parental process without a reset, which yields the desired result for the fBm in a particular case. We then use this result to compute exactly the power spectral density for fBM for all frequency $f$. The asymptotic, large frequency $f$ behaviour of the PSD turns out to be distinctly different for sub- $(H < 1/2)$ and super-diffusive $(H > 1/2)$ fBms. We show that for large $f$, the PSD has a power law tail: $S_T(f) \sim 1/f^{\gamma}$ where the exponent $\gamma= 2H+1$ for $0<H\le 1/2$ (sub-diffusive fBm), while $\gamma= 2$ for all $1/2\le H<1$. Thus, somewhat unexpectedly, the exponent $\gamma=2$ in the superdiffusive case $H>1/2$ sticks to its Brownian value and does not depend on $H$.