Analytic model for transient anomalous diffusion with highly persistent correlations.

Abstract

In recent decades, many stochastic processes have been proposed as models for real world time series data with anomalous spreading, highly persistent correlations, and transient distributional characteristics. We introduce the higher order fractional tempered stable motion as the stochastic integral of the tempered stable motion with respect to a generalized higher order moving average kernel, which provides an analytic model for stochastic processes possessing these characteristics. This stochastic process provides a mathematical model for anomalous diffusion with a transient distribution resembling higher order fractional stable motion on short timescales and higher order fractional Brownian motion in the long run. The specifics of the crossover dynamics from the Lévy stable anomalous diffusion to the Gaussian anomalous diffusion are controlled by explicit parameter values that correspond to physical attributes of the process. It is well suited to modeling anomalous diffusion of any "type" (sub-, super-, regular, or hyperdiffusion) under appropriate parametrizations due to its power-law scaling of variance with respect to time. It is also a useful model for position-velocity-acceleration triples due to its convenient path differentiability and integrability properties. To highlight the potential physical relevance of this model for real world data, we outline its key statistical properties including its covariance structure, memory, and second order self-similarity. We also give an easy to implement elementary method for sample path generation which may be used as a basis for simulation and Monte Carlo studies.