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This study introduces novel Brown and Levy steady-state motions, generalizing Ornstein-Uhlenbeck processes. These versatile stochastic models offer tunable memory and correlation structures, displaying both regular and anomalous features.

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Area of Science:

  • Stochastic processes
  • Mathematical physics

Background:

  • The Ornstein-Uhlenbeck process (OUP) and Levy-driven OUP are fundamental in modeling dynamic systems.
  • Existing models may lack flexibility in capturing complex temporal and amplitude behaviors.

Purpose of the Study:

  • Introduce a novel class of Brown and Levy steady-state motions.
  • Generalize existing Ornstein-Uhlenbeck processes.
  • Explore the properties and applications of these new stochastic motions.

Main Methods:

  • Development of a novel class of stochastic motions based on Brown and Levy processes.
  • Analysis of Markov properties, Langevin dynamics, and steady-state distributions (Gaussian and Levy).
  • Investigation of tunable memory (Joseph effect) and amplitude fluctuation (Noah effect) structures.

Main Results:

  • The novel motions generalize OUP and Levy-driven OUP, exhibiting Markov and Langevin dynamics.
  • Motions can display Noah effect (heavy tails) and Joseph effect (long-range dependence).
  • Two key parameters—critical exponent and clock function—control the Noah and Joseph effects, respectively.

Conclusions:

  • The proposed stochastic model is tractable, versatile, and amenable to analysis.
  • These motions offer a compelling framework for modeling systems with both regular and anomalous features.
  • The tunable nature of memory and correlation structures enhances their applicability in diverse scientific domains.