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We introduce non-Markovian continuous processes with memory, extending the Ornstein-Uhlenbeck process. These processes allow generating stationary stochastic systems with desired correlations.

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Area of Science:

  • Stochastic processes
  • Statistical physics
  • Non-Markovian dynamics

Background:

  • The Mori-Zwanzig equation describes non-Markovian random continuous processes.
  • Markovian Gaussian Ornstein-Uhlenbeck process serves as a foundational model.

Purpose of the Study:

  • To develop a method for constructing non-Markovian continuous stochastic processes with memory.
  • To establish a connection between memory functions and correlation functions.
  • To enable the generation of stationary processes with prescribed correlation properties.

Main Methods:

  • Modification of the Ornstein-Uhlenbeck process by incorporating an integral memory term.
  • Derivation of the higher-order transition probability function and stochastic differential equation for the new processes.
  • Obtaining an equation linking the memory function and the two-point correlation function.
  • Establishing conditions for process stationarity.

Main Results:

  • Proposed processes are continuous-time interpolations of discrete-time higher-order autoregressive sequences.
  • An equation connecting the memory function and the two-point correlation function is derived.
  • A method for generating stationary continuous stochastic processes with a prescribed pair correlation function is presented.
  • Numerical simulations illustrate processes with nonlocal memory.

Conclusions:

  • The developed framework allows for the construction and analysis of non-Markovian processes with memory.
  • The findings provide a method for synthesizing stationary stochastic processes with specific correlation functions.
  • The study offers insights into the behavior of systems with nonlocal memory effects.