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.