The Annals of Probability

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

Loren D. Pitt and Raina S. Robeva

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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.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1338-1376.

First available in Project Euclid: 12 June 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G60: Random fields
Secondary: 31B15: Potentials and capacities, extremal length 31B25: Boundary behavior 60H15: Stochastic partial differential equations [See also 35R60]

Gaussian fields germ fields sharp Markov property spectral synthesis.


Pitt, Loren D.; Robeva, Raina S. On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials. Ann. Probab. 31 (2003), no. 3, 1338--1376. doi:10.1214/aop/1055425783.

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