Anomalous Diffusion: Fractional Brownian Motion vs. Fractional Ito Motion

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.