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 Hurstexponent 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.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

“…In recent years, certain combinations of models were proposed, including switching-diffusivity 11,13,[70][71][72][73][74][75][76][77] and annealed-transienttime models (ATTM), 78 BM with fluctuating or diffusing diffusivity (DD), 70,[79][80][81][82][83][84][85] BM-and anomalous-diffusion-models with ''superstatistically'' distributed model parameters, [86][87][88][89] compound diffusion processes of SBM-DD, 84 SBM-HDPs, 90,91 FBM-DD, 92 FBM-HDPs, 19,93 SBM with exponentially and logarithmically varying D(t), 94 CTRWs with random walks on fractal (RWFs), 95 CTRW-FBM, 16,[96][97][98] as well as several other models, 74,[99][100][101][102][103][104][105][106][107][108][109]…”

“…In recent years, certain combinations of models were proposed, including switching-diffusivity 11,13,70–77 and annealed-transient-time models (ATTM), 78 BM with fluctuating or diffusing diffusivity (DD), 70,79–85 BM- and anomalous-diffusion-models with “superstatistically” distributed model parameters, 86–89 compound diffusion processes of SBM-DD, 84 SBM–HDPs, 90,91 FBM-DD, 92 FBM–HDPs, 19,93 SBM with exponentially and logarithmically varying D ( t ), 94 CTRWs with random walks on fractal (RWFs), 95 CTRW–FBM, 16,96–98 as well as several other models, 74,99–113 including fractional-Langevin-equation (FLE) motion. Renewal processes involving alternation of different types of motions were proposed as well [with, e.g.…”

Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes

“…Having such two similar and comparable models of differing nature should allow building, testing and, tuning indicators attempting to differentiate between the "true" and spurious long-range memory processes. Some of the more recent attempts to discriminate between two different types of memory include [59][60][61][62][63][64][65][66].…”

Selfsimilar stochastic differential equations

“…A detailed discussion of the properties of this density is presented in [1]; in particular, this density has finite variance. Consequently, selfsimilar Ito diffusions display: temporal power-law mean square displacements [29]; and power-law spectral densities -a behavior which was studied in detail in [32][33][34].…”

“…Indeed, selfsimilar motions emerge universally when transcending -via spatio-temporal scaling limits-from the micro level to the macro level. Moreover, selfsimilar motions with finite-variance positions always display the following "anomalous diffusion" behavior [29]: a temporal power-law mean square displacement with exponent 2H.…”