On the wavelet-based simulation of anomalous diffusion

TL;DR

This paper introduces a wavelet-based simulation method for anomalous diffusion models, enabling efficient approximation of complex stochastic processes relevant in physics and microrheology.

Contribution

It presents a novel iterative wavelet-based algorithm that approximates fractional Ornstein-Uhlenbeck and Langevin processes, even when closed-form discretizations are unavailable.

Findings

Algorithm converges quickly to the target process.

Handles cases without closed-form discretizations.

Smoothing procedures improve filter decay.

Abstract

The characterization of particle diffusion is a classical problem in physics and probability theory. The field of microrheology is based on experiments in which microscopic tracer beads are placed into a non-Newtonian fluid and tracked using high speed video capture. The modeling of the behavior of these beads is now an active scientific area which demands multiple stochastic and statistical methods. We propose an approximate wavelet-based simulation technique for two classes of continuous time anomalous diffusion models, the fractional Ornstein-Uhlenbeck process and the fractional generalized Langevin equation. The proposed algorithm is an iterative method that provides approximate discretizations that converge quickly and in an appropriate sense to the continuous time target process. As compared to previous works, it covers cases where the natural discretization of the target process does not have closed form in the time domain. Moreover, we propose smoothing procedures as to speed the time domain decay of the filters.