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Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields
Author(s) -
Biermé Hermine,
Lacaux Céline,
Xiao Yimin
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn122
Subject(s) - mathematics , random field , gaussian random field , brownian motion , hausdorff dimension , gaussian , fractional brownian motion , gaussian process , mathematical analysis , physics , quantum mechanics , statistics
Let X = { X ( t ), t ∈ ℝ N } be a Gaussian random field with values in ℝ d defined by X ( t ) = ( X 1 ( t ), …, X d ( t )), where X 1 , …, X d are independent copies of a centered Gaussian random field X 0 . Under certain general conditions on X 0 , we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X −1 ( F ), where F ⊆ ℝ d is a non‐random Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space‐time white noise and the operator‐scaling Gaussian random fields with stationary increments constructed in [H. Biermé, M. M. Meerschaert and H.‐P. Scheffler, ‘Operator scaling stable random fields’, Stochastic Process. Appl. 117 (2007) 312–332.].