Justification of the Fourier method for hyperbolic equations with random initial conditions

Authors:
Yu. V. Kozachenko and G. I. Slivka

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **69** (2003).

Journal:
Theor. Probability and Math. Statist. **69** (2004), 67-83

MSC (2000):
Primary 60G35; Secondary 35L20

Published electronically:
February 8, 2005

MathSciNet review:
2110906

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Conditions for the existence of a twice differentiable solution of a hyperbolic type partial differential equation with random strongly initial conditions are found in the multidimensional case.

**1.**V. V. Buldygin and Yu. V. Kozachenko,*Metric characterization of random variables and random processes*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716****2.**M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ,*Vypuklye funktsii i prostranstva Orlicha*, Problems of Contemporary Mathematics, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR**0106412****3.**Yu. V. Kozachenko and Yu. A. Koval′chuk,*Boundary value problems with random initial conditions, and functional series from 𝑠𝑢𝑏ᵩ(Ω). I*, Ukraïn. Mat. Zh.**50**(1998), no. 4, 504–515 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**50**(1998), no. 4, 572–585 (1999). MR**1698149**, 10.1007/BF02487389**4.**O. N. Gladkaja,*Conditions for the directional differentiability of sample functions of random fields*, Teor. Verojatnost. i Mat. Statist.**Vyp. 17**(1977), 33–41, 164 (Russian, with English summary). MR**0451375****5.**Yu. Kozachenko, O. Vasylyk, and T. Sottinen,*Path space large deviations of a large buffer with Gaussian input traffic*, Queueing Syst.**42**(2002), no. 2, 113–129. MR**1932123**, 10.1023/A:1020161203956**6.**N. S. Košljakov, È. B. Gliner, and M. M. Smirnov,*Differentsialnye uravneniya matematicheskoi fiziki*, Edited by N. S. Košljakov, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (Russian). MR**0158145****7.**Yu. V. Kozachenko,*Local properties of trajectories of some random functions*, Ukr. Matem. Zh.**17**(1977), 33-40. (Russian)**8.**Ganna Ī. Slivka,*Justification of the application of the Fourier method to the problem of the vibration of a round membrane with random initial conditions*, Vīsn. Kiïv. Unīv. Ser. Fīz.-Mat. Nauki**4**(2002), 31–37 (Ukrainian, with English and Ukrainian summaries). MR**1972945****9.**G. I. Slivka,*A boundary value problem of the mathematical physics with random initial conditions*, Visn. Kyïv Univ. Ser. Fiz.-Mat. Nauk**5**(2002), 172-178. (Ukrainian)

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Additional Information

**Yu. V. Kozachenko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
yvk@univ.kiev.ua

**G. I. Slivka**

Affiliation:
Department of Mathematical Analysis, Faculty for Mathematics, Uzhgorod University, Pidgirna Street 46, Uzhgorod, Ukraine

Email:
aslyvka@yahoo.com

DOI:
https://doi.org/10.1090/S0094-9000-05-00615-0

Received by editor(s):
March 28, 2003

Published electronically:
February 8, 2005

Article copyright:
© Copyright 2005
American Mathematical Society