Fractional heterogeneous telegraph processes: Interplay between heterogeneity, memory, and stochastic resetting

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Telegrapher's equations model random processes with switching states.
• Memory effects and fractional calculus introduce complexities in modeling.
• Stochastic resetting alters the long-term behavior of dynamical systems.

Purpose of the Study:

• To analyze fractional heterogeneous telegraph processes with memory effects.
• To develop methods for obtaining exact solutions for these complex processes.
• To investigate the impact of stochastic resetting on telegraph processes.

Main Methods:

• Integral decomposition method for rigorous analysis.
• Subordination approach to link fractional telegraph equations with Langevin equations.
• Analysis of stochastic resetting within the fractional telegraph framework.

Main Results:

• Exact solutions derived for probability density functions and mean squared displacements.
• Established a relationship between fractional telegraph equations and Langevin equations.
• Obtained exact expressions for nonequilibrium stationary distributions and mean squared displacements under stochastic resetting.

Conclusions:

• The integral decomposition method effectively handles fractional telegraph processes with memory.
• The subordination approach provides a unified framework for related stochastic processes.
• Stochastic resetting leads to exact, analytically tractable nonequilibrium states in these systems.