Two-parameter Garsia–Rodemich–Rumsey inequality and its application to fractional Brownian fields
Author:
K. V. Ral’chenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 75 (2007), 167-178
MSC (2000):
Primary 26D15, 60G15, 60G60
DOI:
https://doi.org/10.1090/S0094-9000-08-00723-0
Published electronically:
January 25, 2008
MathSciNet review:
2321190
Full-text PDF Free Access
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Abstract: The paper contains a generalization of the Garsia–Rodemich–Rumsey inequality to the case of a function of two arguments. Based on this result, we obtain two other inequalities for the fractional Brownian field on the plane, namely an inequality for the Hölder constant of the field and similar bounds for its fractional derivatives.
References
- A. M. Garsia, E. Rodemich, and H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970/71), 565–578. MR 267632, DOI https://doi.org/10.1512/iumj.1970.20.20046
- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- S. A. Īl′chenko and Yu. S. Mīshura, Generalized two-parameter Lebesgue-Stieltjes integrals and their applications to fractional Brownian fields, Ukraïn. Mat. Zh. 56 (2004), no. 4, 435–450 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 56 (2004), no. 4, 527–546. MR 2105898, DOI https://doi.org/10.1007/s11253-005-0065-2
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
- Anna Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85–98. MR 1407935
References
- A. M. Garsia, E. Rodemich, and H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970), no. 6, 565–578. MR 0267632 (42:2534)
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- S. A. Il’chenko and Yu. S. Mishura, Generalized two-parameter Lebesgue–Stieltjes integrals and their applications to fractional Brownian fields, Ukrain. Mat. Zh. 56 (2004), no. 4, 435–450; English transl. in Ukrainian Math. J. 56 (2004), no. 4, 527–546. MR 2105898 (2005i:60068)
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, “Nauka i tekhnika”, Minsk, 1987; English transl., Gordon and Breach Science Publishers, New York, 1993. MR 1347689 (96d:26012)
- A. Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85–98. MR 1407935 (98a:60064)
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Additional Information
K. V. Ral’chenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kyiv 03127, Ukraine
Email:
kr2004@ukr.net
Keywords:
Garsia–Rodemich–Rumsey inequality,
two-parameter stochastic processes,
fractional Brownian fields,
Hölder constant
Received by editor(s):
July 4, 2005
Published electronically:
January 25, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
