Discrete-space and -time analogue of a super-diffusive fractional Brownian motion

TL;DR

This paper develops algorithms to construct lattice and integer-time models of super-diffusive fractional Brownian motion, validated by simulations, highlighting key differences from sub-diffusive fBm.

Contribution

The paper introduces two algorithms for creating lattice and integer-time analogues of super-diffusive fBm, validated through extensive numerical simulations.

Findings

Lattice models replicate the power-law covariance of super-diffusive fBm.

Simulated trajectories match those of super-diffusive fBm.

Constructing sub-diffusive fBm lattice models remains an open challenge.

Abstract

We discuss how to construct reliably well "a lattice and an integer time" version of a super-diffusive continuous-space and -time fractional Brownian motion (fBm) -- an experimentally-relevant non-Markovian Gaussian stochastic process with an everlasting power-law memory on the time-evolution of thermal noises extending over the entire past. We propose two algorithms, which are both validated by extensive numerical simulations showing that the ensuing lattice random walks have not only the same power-law covariance function as the standard fBm, but also individual trajectories follow those of the super-diffusive fBm. Finding a lattice and an integer time analogue of a sub-diffusion fBm, which is an anti-persistent process, remains a challenging open problem. Our results also clarify the relevant difference between sub-diffusive and super-diffusive fBm, that are frequently seen as two very analogous realizations of processes with memory. They are indeed substantially different.