The Annals of Probability

Polarity of points for Gaussian random fields

Robert C. Dalang, Carl Mueller, and Yimin Xiao

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We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions $k\geq1$. Our approach builds on a delicate covering argument developed by M. Talagrand [Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.

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Ann. Probab., Volume 45, Number 6B (2017), 4700-4751.

Received: May 2015
Revised: November 2016
First available in Project Euclid: 12 December 2017

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60G60: Random fields

Hitting probabilities polarity of points critical dimension harmonizable representation stochastic partial differential equations


Dalang, Robert C.; Mueller, Carl; Xiao, Yimin. Polarity of points for Gaussian random fields. Ann. Probab. 45 (2017), no. 6B, 4700--4751. doi:10.1214/17-AOP1176.

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