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

Abstract

Let $\Phi = \{\phi(x)\dvtx x\in \mathbb{R}^2\}$ be a Gaussian random field on the plane. For $A \subset \R^2$, we investigate the relationship between the $\sigma$-field ${\mathcal F}(\Phi, A) = \sigma \{ \phi(x)\dvtx x \in A \} $ and the infinitesimal or germ $\sigma$-field $\,\bigcap_{\varepsilon >0} {\mathcal F} (\Phi, A_{\varepsilon }),$ where $A_{\varepsilon}$ is an $\varepsilon$-neighborhood of A. General analytic conditions are developed giving necessary and sufficient conditions for the equality of these two $\sigma$-fields. These conditions are potential theoretic in nature and are formulated in terms of the reproducing kernel Hilbert space associated with $\Phi $. The Bessel fields $\Phi_{\beta}$\vspace*{-1pt} satisfying the pseudo-partial differential equation $(I-\Delta)^{\beta/2}\phi(x)=\dot W(x)$, $\beta>1$, for which the reproducing kernel Hilbert spaces are identified as spaces of Bessel potentials ${\mathcal L}^{\beta, 2}$, are studied in detail and the conditions for equality are conditions for spectral synthesis in ${\mathcal L}^{\beta,2}$. The case $\beta = 2$ 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.