L 2 Theory of Invariant Random Fields

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We consider a random field X(t) as a function taking values in the Hilbert space of all centred finite-dimensional random vectors with finite expectation of the square of the norm. Examples of inversion formulae, which reconstruct the random measures Z when X is known, are given in Sect. 3.1. Linear differential equations whose right-hand side is a random field, are considered in Sect. 3.2. In Sect. 3.3 we introduce rigged Hilbert spaces of ordinary, smooth, and generalised functions, linear functionals, and spectral characteristics connected to the random field X(t). Using Paley–Wiener theorems, we study how smooth our “smooth functions” are. Bibliographical remarks conclude.