Anomalous Diffusion: Fractional Brownian Motion vs. Fractional Ito Motion

TL;DR

This paper compares fractional Brownian motion and fractional Ito motion as models for anomalous diffusion, highlighting FIM's advantages in simulation and analysis due to its Markovian and martingale properties.

Contribution

It introduces fractional Ito motion as a new, analytically tractable, non-Gaussian model for anomalous diffusion with properties distinct from FBM.

Findings

FIM is a Markov process and a martingale.

FIM is easier to simulate and analyze than FBM.

FIM exhibits non-Gaussian dissipation patterns.

Abstract

Generalizing Brownian motion (BM), fractional Brownian motion (FBM) is a paradigmatic selfsimilar model for anomalous diffusion. Specifically, varying its Hurst exponent, FBM spans: sub-diffusion, regular diffusion, and super-diffusion. As BM, also FBM is a symmetric and Gaussian process, with a continuous trajectory, and with a stationary velocity. In contrast to BM, FBM is neither a Markov process nor a martingale, and its velocity is correlated. Based on a recent study of selfsimilar Ito diffusions, we explore an alternative selfsimilar model for anomalous diffusion: fractional Ito motion (FIM). The FIM model exhibits the same Hurst-exponent behavior as FBM, and it is also a symmetric process with a continuous trajectory. In sharp contrast to FBM, we show that FIM: is not a Gaussian process; is a Markov process; is a martingale; and its velocity is not stationary and is not correlated. On the one hand, FBM is hard to simulate, its analytic tractability is limited, and it generates only a Gaussian dissipation pattern. On the other hand, FIM is easy to simulate, it is analytically tractable, and it generates non-Gaussian dissipation patterns. Moreover, we show that FIM has an intimate linkage to diffusion in a logarithmic potential. With its compelling properties, FIM offers researchers and practitioners a highly workable analytic model for anomalous diffusion.