Stochastic equation for a jumping process with long-time correlations

Area of Science:

• Physics
• Stochastic Processes
• Nonlinear Dynamics

Background:

• 1/f noise is prevalent in various physical systems.
• Generalized Langevin equations describe systems with memory effects.
• Understanding stochastic processes is crucial for modeling complex phenomena.

Purpose of the Study:

• To present a novel jumping process model.
• To analyze its properties, including autocorrelation and relaxation.
• To explore its application as a model for 1/f noise and in generalized Langevin equations.

Main Methods:

• Defining a jumping process with value-dependent rates.
• Analyzing Markovian and stationary properties.
• Characterizing the autocorrelation function as a power law.
• Solving the generalized Langevin equation for specific noise correlations.

Main Results:

• The jumping process exhibits a power-law autocorrelation function.
• It serves as a model for 1/f noise.
• The generalized Langevin equation solution shows sharply falling velocity distribution tails.
• The system demonstrates long-term memory of initial conditions.

Conclusions:

• The proposed jumping process is a versatile model for 1/f noise.
• It provides insights into stochastic forces in generalized Langevin equations.
• The model's long-term memory is a significant characteristic.