Stationary systems of Gaussian processes

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, ${\sigma }^2(t)=\mathrm{Var}W(t)$, and $c\in \mathbb{R}$.

Published in at http://dx.doi.org/10.1214/10-AAP686 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Stationary max-stable fields associated to negative definite functions
• Characterization of invariant measures at the leading edge for competing particle systems
• On the structure of quasi-stationary competing particle systems

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