Memory equations as reduced Markov processes

TL;DR

This paper demonstrates that many linear memory differential equations can be represented as projections of higher-dimensional Markov processes, enabling new analytical tools and more realistic modeling of memory effects.

Contribution

It provides an explicit construction method for Markov processes corresponding to memory equations, extending analysis techniques and improving modeling accuracy.

Findings

Memory equations can be represented as Markov processes in higher dimensions.

The approach allows calculation of equilibrium and asymptotic behavior.

It offers a new way to approximate degenerate memory equations like delay differential equations.

Abstract

A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we propose an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as a change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realistic modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.