Two-scale renormalization-group classification of diffusive processes.

Area of Science:

• Statistical Physics
• Complex Systems Analysis

Background:

• Stochastic processes are fundamental in modeling natural phenomena.
• Classifying these processes based on temporal behavior is crucial for understanding complex systems.
• Existing methods may struggle with nonstationary increments and infinite moments.

Purpose of the Study:

• To introduce a novel classification scheme for stochastic processes using renormalization-group operators.
• To demonstrate the robustness of this scheme across various conditions, including nonstationary increments and infinite second moments.
• To apply the scheme to classify specific physical and biological processes.

Main Methods:

• Utilizing renormalization-group operators to analyze stochastic processes on two distinct time scales.
• Investigating the long-time and short-time behavior through repeated operator application.
• Examining the fixed points of these operators for process subclassification.
• Applying the developed scheme to models of advection-diffusion and bronchial tree diffusion.

Main Results:

• The renormalization-group operator approach effectively classifies stochastic processes based on their temporal dynamics.
• The classification scheme is robust, handling nonstationary increments and infinite second moments.
• Fixed points of the operators provide a means for further subclassification when applicable.
• Successful classification of advection-diffusion and a human bronchial tree diffusion model was achieved.

Conclusions:

• Renormalization-group operators offer a powerful and robust framework for classifying stochastic processes.
• This method enhances the understanding of both short-term and long-term behaviors in complex systems.
• The approach has broad applicability, demonstrated by its use in physical and biological modeling.