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Multiplicative Lévy processes: Itô versus Stratonovich interpretation.

Tomasz Srokowski1

  • 1Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 7, 2010
PubMed
Summary
This summary is machine-generated.

This study analyzes the Langevin equation with multiplicative Lévy noise. The Stratonovich interpretation reveals subdiffusion, unlike the Itô interpretation which shows infinite variance.

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Area of Science:

  • Stochastic processes
  • Statistical physics

Background:

  • The Langevin equation describes systems influenced by random forces.
  • Multiplicative noise, where the random force depends on the system's state, presents unique challenges.
  • Lévy distributions model processes with heavy tails, deviating from Gaussian assumptions.

Purpose of the Study:

  • To investigate the behavior of a Langevin equation driven by multiplicative noise with Lévy distribution and power-law intensity.
  • To derive and solve the corresponding Fokker-Planck equations under both Itô and Stratonovich interpretations.
  • To analyze the system's variance and diffusion properties, comparing analytical and numerical results.

Main Methods:

  • Formulation of the Langevin equation with multiplicative Lévy noise.
  • Derivation of corresponding Fokker-Planck equations for Itô and Stratonovich interpretations.
  • Analytical solution for cases without drift and with a harmonic oscillator potential.
  • Evaluation of the variance and comparison with numerical simulations.

Main Results:

  • The Itô interpretation consistently yields infinite variance.
  • The Stratonovich interpretation can result in finite variance, indicating subdiffusion.
  • Subdiffusion is characterized by variance increasing slower than linearly with time.
  • Analytical predictions align well with numerical simulation outcomes.

Conclusions:

  • The interpretation of stochastic calculus (Itô vs. Stratonovich) significantly impacts the predicted behavior of systems with multiplicative Lévy noise.
  • Subdiffusion, a slower-than-linear increase in variance, is a key characteristic observed under the Stratonovich interpretation.
  • This work provides a theoretical framework and numerical validation for understanding complex stochastic systems.