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We present a new analytical method for solving memory functions in random processes. This method reveals two distinct behaviors for the pair correlation function in stationary telegraph processes.

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Area of Science:

  • * Stochastic processes
  • * Statistical physics
  • * Mathematical modeling

Background:

  • * Random dichotomous processes with memory are crucial in modeling complex systems.
  • * Nonlocal memory effects, where the future depends on a weighted integral of the past, present unique analytical challenges.

Purpose of the Study:

  • * To develop an analytical method for solving integro-differential equations governing processes with memory.
  • * To investigate the stationarity conditions for a telegraph process with exponential memory.
  • * To characterize the behavior of the pair correlation function in such processes.

Main Methods:

  • * Development of a novel analytical closed-form solution for integro-differential equations.
  • * Analysis of stationarity conditions for specific memory functions (exponential).
  • * Derivation and analysis of the pair correlation function.

Main Results:

  • * An analytical closed-form solution is established for the relationship between memory and pair correlation functions.
  • * Stationarity conditions for the exponential memory telegraph process are determined.
  • * The pair correlation function exhibits two distinct forms: exponential decay or damped oscillation.

Conclusions:

  • * The presented method offers a powerful tool for analyzing stochastic processes with memory.
  • * The findings provide a deeper understanding of the statistical properties of telegraph processes.
  • * The dual nature of the pair correlation function highlights the complex dynamics introduced by memory.