Authors’ summary: Let \(\Phi=\{\varphi(x):x\in\mathbb{R}^2\}\) be a Gaussian random field on the plane. For \(A\subset\mathbb{R}^2\), we investigate the relationship between the \(\sigma\)-field \(\mathcal{F}(\Phi,A)=\sigma\{\varphi(x): x\in A\}\) 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 \(\Phi_\beta\) satisfying the pseudo-partial differential equation \((I-\Delta)^{\beta/2}=\dot{W}(x)\), \(\beta>1\), 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 R. C. Dalang and J. B. Walsh [Acta Math. 168, 153–218 (1992; Zbl 0759.60056)] on the Brownian sheet.