Stationary systems of Gaussian processes

Abstract

We describe all countable particle systems on 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 ξ. 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 ξ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and ξ 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 αe^−λx, where α > 0, λ ∈ ℝ, whereas the process ξ is of the form ξ(t) = W(t) − λσ²(t) / 2 + c, where W is a zero-mean Gaussian process with stationary increments, σ²(t) = Var W(t), and c ∈ ℝ.