Journal of Theoretical Probability https://doi.org/10.1007/s10959-019-00964-3 Hausdorff Measure of the Range of Space–Time Anisotropic Gaussian Random Fields Wenqing Ni 1,2 · Zhenlong Chen 1 Received: 10 April 2019 / Revised: 30 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract Let X ={ X (t ) ∈ R d , t ∈ R N } be a centered space–time anisotropic Gaussian random field with stationary increments, whose components are independent but may not be identically distributed. Under certain mild conditions, we determine the exact Hausdorff measure function for the range X ([0, 1] N ). Our result extends those in Talagrand (Ann Probab 23:767–775, 1995) for fractional Brownian motion and Luan and Xiao (J Fourier Anal Appl 18:118–145, 2012) for time-anisotropic and space- isotropic Gaussian random fields. Keywords Space–time anisotropic Gaussian random fields · Strong local nondeterminism · Range · Hausdorff measure Mathematics Subject Classification (2010) 60G15 · 60G17 · 60G60 1 Introduction In probability and statistics there has been of considerable interest in studying Gaussian random fields due to their applications in various scientific areas including physics, engineering, hydrology, biology, economics and finance. Two important examples of This work was supported by the National Natural Science Foundation of China (Grant No. 11971432) and the Education and Scientific Research Foundation for Young and Middle-aged teachers of Fujian Province, China (No. JAT170334). B Zhenlong Chen zlchen@zjsu.edu.cn Wenqing Ni wqni@jmu.edu.cn 1 School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China 2 School of Science, Jimei University, Xiamen 361021, China 123