Spectral content of fractional Brownian motion with stochastic reset

Satya N Majumdar, Gleb Oshanin

Published 2018-06-09Version 1

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 $H$.