First Passage Behavior of Multi-Dimensional Fractional Brownian Motion and Application to Reaction Phenomena

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.