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July 2003 On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials
Loren D. Pitt, Raina S. Robeva
Ann. Probab. 31(3): 1338-1376 (July 2003). DOI: 10.1214/aop/1055425783

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.

Citation

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Loren D. Pitt. Raina S. Robeva. "On the sharp Markov property for Gaussian random fields and spectral synthesis in spaces of Bessel potentials." Ann. Probab. 31 (3) 1338 - 1376, July 2003. https://doi.org/10.1214/aop/1055425783

Information

Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1040.60042
MathSciNet: MR1989436
Digital Object Identifier: 10.1214/aop/1055425783

Subjects:
Primary: 60G15, 60G60
Secondary: 31B15, 31B25, 60H15

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 3 • July 2003
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