Stationary systems of Gaussian processes

TL;DR

This paper classifies all stationary Gaussian particle systems on the real line, characterized by independence, Gaussianity, and stationarity, revealing three distinct families based on the initial measure and the Gaussian process law.

Contribution

It provides a complete classification of stationary Gaussian particle systems, identifying three families with explicit conditions on initial measure and process structure.

Findings

Three families of stationary Gaussian systems identified

Explicit forms of initial measure and Gaussian process provided

Classification covers trivial, linear drift, and exponential density cases

Abstract

We describe all countable particle systems on $\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process $\xi$. We classify all pairs $(\mathfrak{m},\xi)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\mathfrak{m}$ is arbitrary, whereas the process $\xi$ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and $\xi$ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\mathfrak{m}$ has a density of the form $\alpha e^{-\lambda x}$, where $\alpha >0$, $\lambda\in\mathbb{R}$, whereas the process $\xi$ is of the form $\xi(t)=W(t)-\lambda\sigma ^2(t)/2+c$, where $W$ is a zero-mean Gaussian process with stationary increments, $\sigma ^2(t)=\operatorname {Var}W(t)$, and $c\in\mathbb{R}$.