The heat equation with random initial conditions from Orlicz spaces

Authors:
Yu. V. Kozachenko and K. I. Veresh

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **80** (2009).

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

Abstract | References | Similar Articles | Additional Information

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.

**1.**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****2.**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****3.**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****4.**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)**5.**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**, https://doi.org/10.1515/rose.1995.3.3.201**6.**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**, https://doi.org/10.1090/S0094-9000-05-00615-0**7.**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**, https://doi.org/10.1007/s11253-008-0030-y**8.**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

DOI:
https://doi.org/10.1090/S0094-9000-2010-00795-2

Keywords:
Parabolic equations,
Orlicz spaces

Received by editor(s):
February 2, 2009

Published electronically:
August 19, 2010

Article copyright:
© Copyright 2010
American Mathematical Society