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

• 成果类型: Article
• 署名作者: Pitt, LD; Robeva, RS
• 署名单位: University of Virginia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1338-1376
• 关键词: integrals

摘要

Let Phi = {phi(x):x is an element of R-2} be a Gaussian random field on the plane. For A subset of R-2, we investigate the relationship between the sigma-field F(Phi, A) = sigma{phi(x):x is an element of A} and the infinitesimal or germ sigma-field boolean AND(epsilon>0) F(Phi, A(epsilon)), where A(epsilon) is an epsilon-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 (Do satisfying the pseudo-partial differential equation (I - Delta)(beta/2)phi(x) = (W) over dot (x), beta > 1, for which the reproducing kernel Hilbert spaces are identified as spaces of Bessel potentials L-beta,L-2, are studied in detail and the conditions for equality are conditions for spectral synthesis in L-beta,L-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.