Generalized solutions of a hyperbolic equation with a $\varphi$-sub-Gaussian right hand side
Author:
B. V. Dovgaĭ
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 81 (2010), 27-33
MSC (2000):
Primary 60H15, 35L20, 35R60
DOI:
https://doi.org/10.1090/S0094-9000-2010-00807-6
Published electronically:
January 14, 2011
MathSciNet review:
2667307
Full-text PDF Free Access
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Additional Information
Abstract: A hyperbolic equation with a $\varphi$-sub-Gaussian right hand side is considered. Conditions for the existence of a generalized solution of such an equation are found.
References
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- E. Beĭsenbaev and Ju. V. Kozačenko, Uniform convergence in probability of random series, and solutions of boundary value problems with random initial conditions, Teor. Veroyatnost. i Mat. Statist. 21 (1979), 9–23, 163 (Russian, with English summary). MR 550238
- E. Barrasa de la Krus and Yu. V. Kozachenko, Boundary-value problems for equations of mathematical physics with strictly Orlicz random initial conditions, Random Oper. Stochastic Equations 3 (1995), no. 3, 201–220. MR 1354813, DOI https://doi.org/10.1515/rose.1995.3.3.201
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- Yu. V. Kozachenko and B. V. Dovgay, The conditions for application of Fourier method to the solution of nonhomogeneous string oscillation equation with $\phi $-subgaussian right side, Random Oper. Stochastic Equations 13 (2005), no. 3, 281–296. MR 2165325, DOI https://doi.org/10.1163/156939705774286074
- B. V. Dovgay and Yu. V. Kozachenko, Properties of the solution of nonhomogeneous string oscillation equations with $\phi $-subgaussian right side, Random Oper. Stoch. Equ. 17 (2009), no. 3, 221–241. MR 2574102, DOI https://doi.org/10.1515/ROSE.2009.016
- M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
- Yu. V. Kozachenko and G. Ī. Slivka, Justification of the Fourier method for a hyperbolic equation with random initial conditions, Teor. Ĭmovīr. Mat. Stat. 69 (2003), 63–78 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 69 (2004), 67–83 (2005). MR 2110906, DOI https://doi.org/10.1090/S0094-9000-05-00615-0
- Yu. V. Kozachenko and Yu. A. Koval′chuk, Boundary value problems with random initial conditions, and functional series from ${\rm sub}_\phi (\Omega )$. 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, DOI https://doi.org/10.1007/BF02487389
- R. Giuliano Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of $\phi $-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003), 95–124 (English, with English and Italian summaries). MR 2056414
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References
- V. V. Buldygin and Yu. V. Kozachenko, On a question of the applicability of the Fourier method for solving problems with random boundary conditions, Random Processes in Problems of Mathematical Physics, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979, pp. 4–35. (Russian) MR 587658 (83i:35080)
- E. Beisenbaev and Yu. V. Kozachenko, Uniform convergence in probability of random series, and solution of boundary value problems with random initial conditions, Teor. Veroyatnost. i Mat. Statist. 21 (1979), 9–23; English transl. in Theory Probab. Math. Statist. 21 (1980), 9–23. MR 550238 (81g:60036)
- E. Barrasa de la Krus and Yu. V. Kozachenko, Boundary-value problems for equations of mathematical physics with strictly Orlicz random initial conditions, Random Oper. Stochastic Equations 3 (1995), no. 3, 201–220. MR 1354813 (96h:60009)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR 1743716 (2001g:60089)
- B. V. Dovgay and Yu. V. Kozachenko, The condition for application of Fourier method to the solution of nonhomogeneous string oscillation equation with $\varphi$-sub-Gaussian right hand side, Random Oper. Stochastic Equations 13 (2005), no. 3, 281–296. MR 2165325 (2006h:60042)
- B. V. Dovgay and Yu. V. Kozachenko, The properties of solution of nonhomogeneous string oscillation equation with $\varphi$-sub-Gaussian right side, Random Oper. Stochastic Equations 17 (2009), no. 3, 221–241. MR 2574102 (2010k:35281)
- M. A. Krasnosel’skiĭ and J. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958; English transl., P. Noordhoff Ltd., Groningen, 1961. MR 0126722 (23:A4016)
- Yu. V. Kozachenko and G. I. Slivka, Justification of the Fourier method for a hyperbolic equation with random initial conditions, Teor. Imovir. Mat. Stat. 69 (2003), 63–78; English transl. in Theory Probab. Math. Statist. 69 (2004), 67–83. MR 2110906 (2005k:60127)
- Yu. V. Kozachenko and Yu. A. Kovalchuk, Boundary value problems with random initial conditions, and functional series from $\mathrm {Sub}_\varphi (\Omega )$, Ukrain. Mat. Zh. 50 (1998), no. 4, 504–515; English transl. in Ukrainian Math. J. 50 (1999), no. 4, 572–585. MR 1698149 (2000f:60029)
- R. Giuliano Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of $\varphi$-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. XXVII (2003), no. 1, 95–124. MR 2056414 (2005f:60036)
- G. N. Polozhiĭ, Equations of Mathematical Physics, Vysshaya shkola, Moscow, 1964. (Russian)
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Additional Information
B. V. Dovgaĭ
Affiliation:
Department of Operations Research, Faculty for Cybernetics, National Taras Shevchenko University, Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
bogdov@gmail.com
Keywords:
Generalized solutions,
hyperbolic equation,
$\varphi$-sub-Gaussian stochastic process
Received by editor(s):
May 20, 2009
Published electronically:
January 14, 2011
Article copyright:
© Copyright 2010
American Mathematical Society