Analytic model for transient anomalous diffusion with highly persistent correlations

Area of Science:

• Stochastic processes
• Anomalous diffusion modeling
• Mathematical physics

Background:

• Real-world time series often exhibit anomalous spreading, persistent correlations, and transient distributions.
• Existing stochastic models may not fully capture these complex characteristics.
• There is a need for analytic models that can describe these phenomena.

Purpose of the Study:

• To introduce the higher order fractional tempered stable motion as a novel analytic model.
• To provide a mathematical framework for anomalous diffusion with transient characteristics.
• To demonstrate its suitability for modeling diverse diffusion types and physical processes.

Main Methods:

• Defining the higher order fractional tempered stable motion as a stochastic integral.
• Analyzing its properties, including covariance structure, memory, and self-similarity.
• Developing an elementary method for sample path generation.

Main Results:

• The proposed process models anomalous diffusion, transitioning from higher order fractional stable motion to higher order fractional Brownian motion.
• Crossover dynamics between Lévy stable and Gaussian anomalous diffusion are parameter-controlled.
• The model accommodates sub-, super-, regular, and hyperdiffusion based on parametrization.
• It is suitable for modeling position-velocity-acceleration triples due to path properties.

Conclusions:

• Higher order fractional tempered stable motion is a versatile analytic model for complex time series.
• Its parameters allow control over diffusion dynamics and physical attributes.
• The model's properties and generation method support simulation and real-world data analysis.