# Archive ouverte HAL – Spectral content of fractional Brownian motion with stochastic reset

### Satya N Majumdar 1 Gleb Oshanin 2 Satya Majumdar 1

#### Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2018, 51 (43), 〈10.1088/1751-8121/aadef0〉

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

• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
• 2. LPTMC - Laboratoire de Physique Théorique de la Matière Condensée