[Paper Review] On the Fractional Riemann-Liouville Integral of Gauss-Markov processes and applications

Mario Abundo, Enrica Pirozzi|arXiv (Cornell University)|May 20, 2019

Statistical Mechanics and Entropy33 references4 citations

TL;DR

This paper investigates the fractional Riemann-Liouville integral of Gauss-Markov processes, deriving closed-form expressions for the mean, variance, and covariance of the resulting processes. It demonstrates that the fractional order parameter α significantly modulates memory effects and correlation structure, with numerical results showing that FIBM covariance increases rapidly with α, while FISOU exhibits broader correlation patterns than FIOU due to its random initial condition, enabling enhanced modeling of long-range memory in neuronal dynamics.

ABSTRACT

We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order $α\in (0,1)$ of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central rule, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter $α$ affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractional integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

Motivation & Objective

• To extend the theory of integrated Gauss-Markov processes by replacing ordinary Riemann integrals with fractional Riemann-Liouville integrals of order α ∈ (0,1).
• To derive analytical expressions for the mean, variance, and covariance functions of the fractional integral of standard Brownian motion and non-stationary/stationary Ornstein-Uhlenbeck processes.
• To investigate how the fractional order α affects the memory and correlation structure of the resulting processes through numerical and graphical comparisons.
• To provide a foundation for simulating sample paths and analyzing first-passage times in fractional neuronal models with long-range memory.
• To propose a neuronal model based on fractional-integrated correlated input processes, leveraging the enhanced memory control offered by α.

Proposed method

• The fractional Riemann-Liouville integral of order α is applied to Gauss-Markov processes, including standard Brownian motion and both non-stationary and stationary Ornstein-Uhlenbeck processes.
• Closed-form expressions for the mean, variance, and covariance functions are derived using properties of fractional calculus and the second-order structure of Gauss-Markov processes.
• Numerical evaluations of the covariance functions are performed for varying α to analyze the impact of the fractional order on memory and correlation dynamics.
• Graphical comparisons are conducted using 3D plots and color maps to visualize the evolution of covariance functions across time and α values.
• The simulation of sample paths is enabled by the derived covariance functions, supporting further analysis of first-passage times.
• A neuronal model is proposed where the input process is modeled as the fractional integral of a correlated Gauss-Markov process, with α serving as a tunable memory parameter.

Experimental results

Research questions

• RQ1 How does replacing the ordinary Riemann integral with the fractional Riemann-Liouville integral of order α affect the statistical properties of Gauss-Markov processes?
• RQ2 What are the closed-form expressions for the mean, variance, and covariance of the fractional integral of standard Brownian motion and Ornstein-Uhlenbeck processes?
• RQ3 How does the fractional order α influence the correlation structure and memory effects in the resulting processes?
• RQ4 What are the differences in covariance behavior between the fractional integral of non-stationary and stationary Ornstein-Uhlenbeck processes?
• RQ5 Can the derived covariance functions enable efficient simulation of sample paths and first-passage time analysis in neuronal models?

Key findings

• The covariance function of the fractional integral of standard Brownian motion (FIBM) increases rapidly with α, reaching values around 15 for α ≈ 1, indicating strong memory effects.
• The covariance of the fractional integral of the non-stationary Ornstein-Uhlenbeck process (FIOU) grows more slowly, reaching approximately 2 for α ≈ 1, reflecting weaker correlation.
• The fractional integral of the stationary Ornstein-Uhlenbeck process (FISOU) exhibits slightly higher covariance values than FIOU across all α, with more diffuse correlation patterns, especially for small t and u near the diagonal.
• For α = 0.2, FISOU covariance shows higher values for small t and u close to the diagonal, while for α = 0.5 and 0.8, the FISOU and FIOU covariance patterns become increasingly similar, though FISOU maintains broader correlation.
• The variance and covariance of all three processes increase with α for large t, as expected, confirming the role of α in enhancing memory persistence.
• The graphical and numerical comparisons confirm that α is a powerful tuning parameter for controlling memory length and time-scale dynamics in complex systems, particularly in neuronal models.