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

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

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*, Amer. Math. Soc., Providence, RI, 2000. MR**1743716 (2001g:60089)****2.**M. A. Krasnosel'ski and Ya. B. Ruticki,*Convex Functions and Orlicz Spaces*, Fizmatgiz, Moscow, 1958; English transl., Noordhof, Gröningen, 1961. MR**0106412 (21:5144)****3.**Yu. V. Kozachenko and Yu. A. Koval'chuk,*Boundary value problems with random initial conditions and series of functions of*, Ukr. Matem. Zh.**50**(1998), no. 4, 504-515; English transl. in Ukrainian Math. J.**50**(1998), no. 4, 572-585 (1999). MR**1698149 (2000f:60029)****4.**O. N. Gladkaya,*Conditions for directional differentiability of the sample functions of random fields*, Teor. Veroyatnost. Matem. Statist.**17**(1977), 33-40; English transl. in Theory Probab. Mathem. Statist.**17**(1977), 35-43. MR**0451375 (56:9661)****5.**Yu. V. Kozachenko, O. Vasylyk, and T. Sottinen,*Path space large deviations of a large buffer with Gaussian input traffic*, Queueing System**42**(2002), no. 2, 113-129. MR**1932123 (2004a:60149)****6.**N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov,*Differential Equations of Mathematical Physics*, ``Vysshaya Shkola'', Moscow, 1962; English transl., North-Holland, Amsterdam, 1964. MR**0158145 (28:1371)****7.**Yu. V. Kozachenko,*Local properties of trajectories of some random functions*, Ukr. Matem. Zh.**17**(1977), 33-40. (Russian)**8.**G. I. Slivka,*Justification of the application of the Fourier method to the problem of the vibration of a round membrane with random initial conditions*, Visn. Kyïv Univ. Ser. Fiz.-Mat. Nauk**4**(2002), 31-37. (Ukrainian) 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