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Power Levy motion. I. Diffusion.

Iddo Eliazar1

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Power Levy motion (PLM) is introduced as a generalization of Levy motion, analogous to power Brownian motion (PBM) generalizing Brownian motion. This study constructs PLM, exploring its diffusion properties like anomalous diffusion and aging.

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Area of Science:

  • Stochastic Processes
  • Mathematical Physics
  • Statistical Mechanics

Background:

  • Brownian motion is a fundamental stochastic process, representing the scaling limit of finite-variance random walks.
  • Levy motion, a stable and symmetric Levy process, serves as the infinite-variance counterpart to Brownian motion.
  • Power Brownian motion (PBM) is a recent generalization of Brownian motion exhibiting Markovian properties and anomalous diffusion.

Purpose of the Study:

  • Introduce and construct Power Levy Motion (PLM), a novel stochastic process.
  • Explore the diffusion-related properties of PLM, extending concepts from PBM to the infinite-variance domain.
  • Lay the groundwork for further investigation of PLM from an "evolution perspective" in a companion paper.

Main Methods:

  • Construction of the Power Levy Motion (PLM) process.
  • Analysis of PLM increments and their Fourier structure.
  • Investigation of self-similarity, Hurst exponent, sub-diffusion, super-diffusion, aging, and Holder exponent through a "diffusion perspective".

Main Results:

  • PLM is successfully constructed and its emergence is explained.
  • Key diffusion characteristics of PLM, including anomalous diffusion behaviors (sub- and super-diffusion) and aging/anti-aging properties, are detailed.
  • The Fourier structure of increments and the Holder exponent of PLM are analyzed.

Conclusions:

  • Power Levy Motion (PLM) is established as a significant generalization in the realm of stochastic processes.
  • PLM offers a framework to study anomalous diffusion phenomena in infinite-variance settings.
  • This work provides a comprehensive "diffusion perspective" on PLM, paving the way for future research.