White noises, random fields and pseudo-differential operators

Our aim in this note is to build quite general random fields with correlation distances given by a small parameter ε. It seems to be natural for that purpose to use ε−pseudo-differential operators. We will see how to compute the generalized power spectrum using Wigner measures. Using the previous tools, we will discuss natural statistics of waves, which we call “microcanonical”. 1 White noises Let (H, 〈.|.〉) be an Hilbert space. There exists a canonical Gaussian random field on it called the white noise and denoted by wH (or simply w if there is no possible confusion). This random field is defined by the properties that: • For all ~e ∈ H, E(〈w|~e〉) = 0 • For all ~e, ~ f ∈ H, E(〈w|~e〉〈w|~ f〉) = 〈~e|~ f〉 More concretely, if (~ ei) is an orthonormal basis of H and w = ∑ wi~ ei, we have E(wiwj) = δij and hence E(〈Aw|w〉) = Trace(A). Unfortunately, w is not a random vector in H unless dimH <∞ , but only a random Schwartz distribution. If w were a vector inH, we would have w = ∑ wi~ ei and we see that E(‖w‖) = ∑ E(|wi|) = dimH =∞ . We have nevertheless the following usefull proposition: ∗Institut Fourier, Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martin d’Hères Cedex (France); [email protected]