# Continuous stochastic processes with non-local memory

## TL;DR

This paper introduces a class of non-Markovian continuous stochastic processes with non-local memory, connecting them to higher-order autoregressive sequences, and provides methods for their simulation and analysis.

## Contribution

It develops a framework for non-Markovian processes with memory, linking them to autoregressive models and offering a way to generate processes with specific correlation functions.

## Key findings

- Derived an equation linking memory kernel and correlation function
- Established conditions for process stationarity
- Presented numerical simulations of 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.