Space Representation of Stochastic Processes with Delay

TL;DR

This paper demonstrates that certain delayed stochastic processes can be mapped onto local two-dimensional processes, revealing that their critical behavior is independent of short delays and related to directed percolation properties.

Contribution

It introduces a method to map non-local delayed processes onto local 2D processes, linking their critical behavior to directed percolation universality.

Findings

Mapping of delayed processes to 2D local processes under coprimality condition.

Critical behavior independence from short delay in certain stochastic models.

Autocorrelation functions relate to directed percolation critical properties.

Abstract

We show that a time series $x_t$ evolving by a non-local update rule $x_t = f (x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.