Continuous stochastic processes with non-local memory
Abstract
We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into expression for the higher-order transition probability function and stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with prescribed pair correlation function. As illustration, some examples of numerical simulation of the processes with non-local memory are presented.
Keywords
Cite
@article{arxiv.1904.03514,
title = {Continuous stochastic processes with non-local memory},
author = {S. S. Melnyk and V. A. Yampol'skii and O. V. Usatenko},
journal= {arXiv preprint arXiv:1904.03514},
year = {2019}
}
Comments
7 pages, 2 figures