Goldstein-Kac telegraph processes with random speeds: Path probabilities, likelihoods, and reported Lévy flights

Area of Science:

• Statistical Physics
• Stochastic Processes
• Computational Modeling

Background:

• The Goldstein-Kac telegraph process models particle motion with constant speed and random direction changes.
• Its inherent mathematical properties limit empirical validation from real-world data.
• This process is relevant to various physical phenomena but lacks practical data-fitting capabilities.

Purpose of the Study:

• To address the limitations of the Goldstein-Kac telegraph process for empirical validation.
• To develop a generalized telegraph process with random speeds.
• To enable robust parameter estimation and analysis of particle diffusion models.

Main Methods:

• Introducing random speeds into the Goldstein-Kac telegraph process.
• Regularizing ballistic terms through speed randomization.
• Employing the unscented transform for approximating diffusion components.
• Calculating particle path probabilities and parameter likelihoods computationally.

Main Results:

• The modified process allows for a posteriori empirical validation using data.
• The unscented transform provides an efficient approximation for the diffusion component.
• The generalized model yields computationally efficient and robust probability evaluations.
• Demonstrated non-Gaussian asymptotic spatial distributions, similar to Lévy walkers.

Conclusions:

• Randomizing particle speeds regularizes the Goldstein-Kac telegraph process, enabling empirical validation.
• The generalized model offers a computationally efficient and robust framework for analyzing particle diffusion.
• This approach can replicate complex diffusion behaviors, such as those observed in Lévy walks.