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Bulletin of the London Mathematical Society Advance Access originally published online on February 19, 2009
Bulletin of the London Mathematical Society 2009 41(2):253-273; doi:10.1112/blms/bdn122
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© 2009 London Mathematical Society

Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields

Hermine Biermé

MAP5
Université Paris Descartes
CNRS UMR 8145
45 rue des Saints-Pères
75006 Paris
France
hermine.bierme@mi.parisdescartes.fr
http://www.math-info.univ-paris5.fr/~bierme/

Céline Lacaux

Institut Élie Cartan
Nancy-Université
CNRS, INRIA
Boulevard des Aiguillettes, BP 239
F-54506 Vandoeuvre-Lès-Nancy
France
http://www.iecn.u-nancy.fr/~lacaux/

Yimin Xiao

Department of Statistics and Probability
Michigan State University
A-413 Wells Hall
East Lansing, MI 48824
USA
xiao@stt.msu.edu
http://www.stt.msu.edu/~xiaoyimi/

Received 15 March 2008. Revision received 2 October 2008.

Let X = {X(t), t isin RN} be a Gaussian random field with values in Rd defined by X(t) = (X1(t), ..., Xd(t)), where X1, ..., Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions on X0, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X–1(F), where F {subseteq} Rd 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.].

2000 Mathematics Subject Classification 60G60, 60G15, 60G17, 28A80.

This work was partially supported by ANR grants ANR-05-BLAN-0017 and ANR-06-BLAN-0289. The research of Yimin Xiao is also supported partially by NSF grant DMS-0706728.


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