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Yusuke Uchiyama1, Takanori Kadoya1, Hidetoshi Konno2

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This study introduces a fractional generalized Cauchy process (FGCP) to model anomalous fluctuations. The FGCP exhibits intermittent dynamics and weak ergodicity breaking, offering new insights into complex systems.

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Area of Science:

  • * Stochastic processes
  • * Statistical physics
  • * Anomalous diffusion

Background:

  • * Anomalous fluctuations are prevalent in complex systems.
  • * Existing models may not fully capture intermittent dynamics.
  • * Understanding ergodicity breaking is crucial for system analysis.

Purpose of the Study:

  • * Introduce a novel fractional generalized Cauchy process (FGCP).
  • * Describe subordinated anomalous fluctuations using FGCP.
  • * Analyze the intermittent dynamics and ergodicity of the FGCP.

Main Methods:

  • * Employing additive and multiplicative Gaussian white noise.
  • * Utilizing Ito stochastic integral for analytical representation.
  • * Applying fractional Feynman-Kac formula for ergodicity analysis.

Main Results:

  • * FGCP demonstrates intermittent dynamics on random time durations.
  • * The stationary state probability density function follows a generalized Cauchy distribution.
  • * Weak ergodicity breaking is linked to subordinator existence and/or variance divergence.

Conclusions:

  • * The FGCP provides a robust framework for modeling anomalous fluctuations.
  • * Intermittent dynamics and weak ergodicity breaking are key features of the FGCP.
  • * The findings offer a deeper understanding of complex system behaviors.