Visitation Dynamics of $d$-Dimensional Fractional Brownian Motion

TL;DR

This paper investigates the visitation dynamics of d-dimensional fractional Brownian motion, revealing the temporal regimes of site visitation times through scaling arguments, simulations, and experimental data analysis.

Contribution

It provides a comprehensive analysis of the visitation time distribution for d-dimensional fBm, combining theoretical scaling, numerical simulations, and real-world biological data.

Findings

Identified all temporal regimes of the visitation time distribution.

Validated theoretical predictions with extensive Monte Carlo simulations.

Applied results to biological tracking data of telomeres, microspheres, and vacuoles.

Abstract

The fractional Brownian motion (fBm) is a paradigmatic strongly non-Markovian process with broad applications in various fields. Despite their importance, the properties of the territory covered by a $d$-dimensional fBm have remained elusive so far. Here, we study the visitation dynamics of the fBm by considering the time $\tau_n$ required to visit a site, defined as a unit cell of a $d$-dimensional lattice, when $n$ sites have been visited. Relying on scaling arguments, we determine all temporal regimes of the probability distribution function of $\tau_n$. These results are confirmed by extensive numerical simulations that employ large-deviation Monte Carlo algorithms. Besides these theoretical aspects, our results account for the tracking data of telomeres in the nucleus of mammalian cells, microspheres in an agorose gel, and vacuoles in the amoeba, which are experimental realizations of fBm.