Stationary systems of Gaussian processes (original) (raw)

2010, The Annals of Applied Probability

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 m and moving independently of each other according to the law of some Gaussian process ξ. We classify all pairs (m, ξ) generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure m is arbitrary, whereas the process ξ is stationary. In the second family, the measure 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 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) − λσ 2 (t)/2 + c, where W is a zero-mean Gaussian process with stationary increments, σ 2 (t) = Var W (t), and c ∈ R. . This reprint differs from the original in pagination and typographic detail. 1 2 Z. KABLUCHKO 4 Z. KABLUCHKO 2. The set S 2 consists of all pairs (m, ξ), where m = αe 0 and {ξ