Abstract
Let \(X= \{X(t) \in \mathbb{R}^d, t\in \mathbb{R}^N\}\) be a centered space-anisotropic Gaussian random field whose components satisfy some mild conditions. By introducing a new anisotropic metric in ℝd, we obtain the Hausdorff and packing dimension in the new metric for the image of X. Moreover, the Hausdorff dimension in the new metric for the image of X has a uniform version.
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References
Adler, R. J.: The Geometry of random fields, John Wiley and Sons Ltd., New York, 1981
Estrade, A., Wu, D., Xiao, Y.: Packing dimension results for anisotropic Gaussian random fields. Commun. Stoch. Anal., 5, 41–64 (2011)
Falconer, K. J.: Fractal Geometry-Mathematical Foundations and Applications, 2nd edition, John Wiley and Sons Ltd., Chichester, 2003
Falconer, K. J., Howroyd, D.: Packing dimensions for projections and dimension profiles. Math. Proc. Camb. Philos. Soc., 121, 926–286 (1997)
Kaufman, R.: Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris, 268, 727–728 (1968)
Khoshnevisan, D., Wu, D., Xiao, Y.: Sectorial local nondeterminism and the geometry of the Brownian sheet. Electron. J. Probab., 11, 817–843 (2006)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Cambrige University Press, Cambrige, 1995
Monrad, D., Pitt, L. D.: Local nondeterminism and Hausdorff dimension. Progress in Probability and Statistics. Seminar on Stochastic Processes 1986. Cindlar E., Chung K. L., Getoor R. K., eds. Birkhäuser, Boston, 1987, 163–189
Mountford, T. S.: Uniform dimension results for the Brownian sheet. Ann. Probab., 17, 1454–1462 (1989)
Wu, D., Xiao, Y.: Uniform dimension results for Gaussian random fields. Sci. China Ser. A, 52, 1478–1496 (2009)
Xiao, Y.: Dimension results for Gaussian vector fields and index-α stable fields. Ann. Probab., 23, 273–291 (1995)
Xiao, Y.: Packing dimension of the image of fractional Brownian motion. Stat. Probabil. Lett., 33, 379–387 (1997)
Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. A Minicourse on Stochastic Partial Differential Equations, (D. Khoshnevisan and F. Rassoul-Agha, editors), Lecture Notes in Math, 1962, Springer, New York, 2009, 145–212
Xiao, Y.: Recent developments on fractal properties of Gaussian random fields. Further Developments in Fractals and Related Fields. (Julien Barral and Stephane Seuret, editors), Springer, New York, 2013, 255–288
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We cordially thank Professor Xiao for his kindly help and the referees for their time and comments.
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Supported by the Humanities and Social Sciences Research Project of Ministry of Education (Grant No. 18YJA910001), the National Natural Science Foundation of China (Grant No. 11371321), the first author is also supported by the Education and Scientific Research Foundation for Young and Middle-aged teachers of Fujian Province (Grant No. B17154)
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Ni, W.Q., Chen, Z.L. & Wang, W.G. Dimension Results for Space-anisotropic Gaussian Random Fields. Acta. Math. Sin.-English Ser. 35, 391–406 (2019). https://doi.org/10.1007/s10114-018-8016-7
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DOI: https://doi.org/10.1007/s10114-018-8016-7