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Multiplicative Lévy processes: Itô versus Stratonovich interpretation
1Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland.
Summary
This study analyzes the Langevin equation with multiplicative Lévy noise. The Stratonovich interpretation reveals subdiffusion, unlike the Itô interpretation which shows infinite variance.
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.
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