On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials. (English) Zbl 1040.60042

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.


60G60 Random fields
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
60G15 Gaussian processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)


Zbl 0759.60056
Full Text: DOI Euclid


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