First passage behaviour of multi-dimensional fractional Brownian motion and application to reaction phenomena

TL;DR

This paper investigates the first passage properties of multi-dimensional fractional Brownian motion (FBM) and applies these findings to understand reaction phenomena involving FBM in physics and biology.

Contribution

It introduces a multi-dimensional FBM model as a superposition of one-dimensional FBMs and analyzes its first passage behavior, extending existing one-dimensional results.

Findings

Derived asymptotic first passage time statistics for multi-dimensional FBM.

Compared theoretical predictions with simulations showing good agreement.

Applied results to model diffusion-limited reactions in multi-dimensional spaces.

Abstract

Fractional Brownian motion is a generalised Gaussian diffusive process that is found to describe numerous stochastic phenomena in physics and biology. Here we introduce a multi-dimensional fractional Brownian motion (FBM) defined as a superposition of conventional FBM for each coordinate in analogy to multi-dimensional Brownian motion, and study its first passage properties. Starting from the well-established first passage time statistics of one-dimensional FBM and the associated approximation schemes, we explore the first passage time behaviour of multi-dimensional FBM and compare these results with simulations. The asymptotic kinetic behaviour of diffusion-limited reactions of reactant particles performing FBM in a one- and multi-dimensional space is studied based on the corresponding first passage time statistics.