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 m and moving independently of each other according to the law of some Gaussian process xi. We classify all pairs (m, xi) generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure m is arbitrary, whereas the process xi is stationary. In the second family, the measure 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 m has a density of the form alpha e(-lambda x), where alpha > 0, lambda is an element of 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) = VarW(t), and c is an element of R.