The heat equation with random initial conditions from Orlicz spaces
Authors:
Yu. V. Kozachenko and K. I. Veresh
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 80 (2010), 71-84
MSC (2000):
Primary 60G60, 60G17
DOI:
https://doi.org/10.1090/S0094-9000-2010-00795-2
Published electronically:
August 19, 2010
MathSciNet review:
2541953
Full-text PDF Free Access
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Abstract: Conditions for justification of the Fourier method for parabolic equations with random initial conditions from Orlicz spaces of random variables are obtained. Bounds for the distribution of the supremum of solutions of such equations are found.
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 (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979, pp. 4–35, 150 (Russian). MR 587658
- 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
- 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
- B. V. Dovgay, Yu. V. Kozachenko, and G. I. Slyvka-Tylyshchak, Boundary-Value Problems of Mathematical Physics with Random Factors, Kyiv University, Kyiv, 2008. (Ukrainian)
- 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
- 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 M. M. Perestyuk, On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. I, Ukraïn. Mat. Zh. 59 (2007), no. 12, 1647–1660 (Ukrainian, with English and Russian summaries); English transl., Ukrainian Math. J. 59 (2007), no. 12, 1850–1869. MR 2411593, DOI https://doi.org/10.1007/s11253-008-0030-y
- I. G. Polozhyĭ, Equations of Mathematical Physics, Vysshaya Shkola, Moscow, 1964. (Russian)
References
- V. V. Buldygin and Yu. V. Kozachenko, On an applicability of the Fourier method for solving problems with random boundary conditions, Random Processes in the Problems of Mathematical Physics, Institute of Mathematics, Academy of Science of Ukr. SSR, Kiev, 1979, pp. 4–35. (Russian) MR 587658 (83i:35080)
- E. Beĭsenbaev and Yu. V. Kozachenko, Uniform convergence of random series in probability and the solution of boundary-value problems with random initial conditions, Teor. Imovir. Mat. Stat. 21 (1979), 9–23; English transl. in Theory Probab. Math. Statist. 21 (1980), 9–24. MR 550238 (81g:60036)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., Translations of Math. Mono., vol. 188, American Mathematical Society, Providence, RI, 2000. MR 1743716 (2001g:60089)
- B. V. Dovgay, Yu. V. Kozachenko, and G. I. Slyvka-Tylyshchak, Boundary-Value Problems of Mathematical Physics with Random Factors, Kyiv University, Kyiv, 2008. (Ukrainian)
- 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. Stoch. Eq. 3 (1995), no. 3, 201–220. MR 1354813 (96h:60009)
- Yu. V. Kozachenko and G. I. Slivka, Justification of the Fourier method for hyperbolic equations 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 M. M. Perestyuk, On the uniform convergence of wavelet expansions of random processes belonging to the Orlicz spaces of random variables. I, Ukr. Matem. Zh. 59 (2007), no. 12, 1647–1659; English transl. in Ukrain. Math. J. 59 (2007), no. 12, 1850–1869. MR 2411593 (2009b:60058)
- I. G. Polozhyĭ, Equations of Mathematical Physics, Vysshaya Shkola, Moscow, 1964. (Russian)
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Additional Information
Yu. V. Kozachenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kiev 03127, Ukraine
Email:
yvk@univ.kiev.ua
K. I. Veresh
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kiev 03127, Ukraine
Email:
yvk@univ.kiev.ua
Keywords:
Parabolic equations,
Orlicz spaces
Received by editor(s):
February 2, 2009
Published electronically:
August 19, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
