On the wavelet-based simulation of anomalous diffusion

Abstract

The field of microrheology is based on experiments involving particle diffusion. Microscopic tracer beads are placed into a non-Newtonian fluid and tracked using high speed video capture and light microscopy. The modelling of the behaviour 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 to minimize the border effect via smoothing.

Keywords:

• simulation
• wavelets
• anomalous diffusion
• generalized Langevin equation
• fractional Ornstein–Uhlenbeck process

AMS Subject Classification::

• 60G10
• 60H35
• 42C40
• 60G15

Acknowledgements

The authors would like to thank Vladas Pipiras for his comments on this work.