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Two-parameter Garsia-Rodemich-Rumsey inequality and its application to fractional Brownian fields
Author(s):
K.
V.
Ral'chenko
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 75
(2006).
Journal:
Theor. Probability and Math. Statist.
No. 75
(2007),
167-178.
MSC (2000):
Primary 26D15, 60G15, 60G60
Posted:
January 25, 2008
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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:
-
- 1.
- 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)
- 2.
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55-81. MR 1893308 (2003f:60105)
- 3.
- 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)
- 4.
- 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)
- 5.
- 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
DOI:
10.1090/S0094-9000-08-00723-0
PII:
S 0094-9000(08)00723-0
Keywords:
Garsia--Rodemich--Rumsey inequality,
two-parameter stochastic processes,
fractional Brownian fields,
H\"older constant
Received by editor(s):
4/JUL/2005
Posted:
January 25, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
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