On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials

Abstract

Let Φ={ϕ(x)\dvtxx∈R2} be a Gaussian random field on the plane. For A⊂\R2, we investigate the relationship between the σ-field F(Φ,A)=σ{ϕ(x)\dvtxx∈A} and the infinitesimal or germ σ-field ⋂e>0F(Φ,Ae), where Ae is an e-neighborhood of A. General analytic conditions are developed giving necessary and sufficient conditions for the equality of these two σ-fields. These conditions are potential theoretic in nature and are formulated in terms of the reproducing kernel Hilbert space associated with Φ. The Bessel fields Φβ\vspace*{-1pt} satisfying the pseudo-partial differential equation (I−Δ)β/2ϕ(x)=W˙(x), β>1, for which the reproducing kernel Hilbert spaces are identified as spaces of Bessel potentials Lβ,2, are studied in detail and the conditions for equality are conditions for spectral synthesis in Lβ,2. The case β=2 is of special interest, and we deduce sharp conditions for the sharp Markov property to hold here, complementing the work of Dalang and Walsh on the Brownian sheet.