The heat equation on line with random right part from Orlicz space. (English) Zbl 1325.35072
Summary: In this paper, the heat equation with random right part is examined. In particular, we give conditions for existence with probability one of the solutions in the case when the right part is a random field, sample continuous with probability one from the Orlicz space. Estimation for the distribution of the supremum of solutions of such equations is found.
References:
| [1] | Angulo J.M., Ruiz-Medina M.D., Anh V.V., Grecksch W. Fractional diffusion and fractional heat equation. Adv. in Appl. Probab. 2000, 32 (4), 1077-1099. doi: 10.1239/aap/1013540349 · Zbl 0986.60077 · doi:10.1239/aap/1013540349 |
| [2] | Beghin L., Kozachenko Yu., Orsingher E., Sakhno L. On the solution of linear odd-order heat-type equations with random initial conditions. J. Stat. Phys. 2007, 127 (4), 721-739. doi: 10.1007/s10955-007-9309-x · Zbl 1147.82328 · doi:10.1007/s10955-007-9309-x |
| [3] | Barrasa de La Krus E., Kozachenko Yu.V. Boundary-value problems for equations of mathematical physics with strictly Orlicz random initial conditions. Random Oper. and Stoch. Equ. 1995, 3 (3), 201-220. · Zbl 0836.35165 |
| [4] | Bejsenbaev E., Kozachenko Yu.V. Uniform convergence in probability of random series, and solutions of boundary value problems with random initial conditions. Teor. Imovir. Mat. Stat. 1979, 21, 9-23. (in Russian) · Zbl 0486.60006 |
| [5] | Buldygin V.V., Kozachenko Yu.V. Metric Characterization of Random Variables and Random processes. AMS, Providence, Rhode Island, 2000. · Zbl 0998.60503 |
| [6] | Markovich B.M. Equations of Mathematical Physics. Lviv Polytechnic Publishing House, Lviv, 2010. (in Ukrainian) |
| [7] | Kozachenko Yu.V., Leonenko G.M. Extremal behavior of the heat random field. Extremes 2005, 8 (3), 191-205. doi: 10.1007/s10687-006-7967-8 · Zbl 1115.60054 · doi:10.1007/s10687-006-7967-8 |
| [8] | Kozachenko Yu.V., Slyvka G.I. Justification of the Fourier method for hyperbolic equations with random initial conditions. Theory Probab. and Math. Statist. 2004, 69, 67-83. doi: 10.1090/S0094-9000-05-00615-0 (translation of Teor. Imovir. Mat. Stat. 2003, 69, 63-78. (in Russian)) · doi:10.1090/S0094-9000-05-00615-0 |
| [9] | Kozachenko Yu.V., Slyvka G.I. Modelling a solution of a hyperbolic equation with random initial conditions. Theory Probab. Math. Statist. 2007, 74, 59-75. doi: 10.1090/S0094-9000-07-00698-9 (translation of Teor. Imovir. Mat. Stat. 2006, 74, 52-67 (in Russian)) · doi:10.1090/S0094-9000-07-00698-9 |
| [10] | Kozachenko Yu.V., Slyvka-Tylyshchak A.I. The Cauchy problem for the heat equation with a random right side. Random Oper. Stoch. Equ. 2014, 22 (1), 53-64. doi: 10.1515/rose-2014-0006 · Zbl 1284.35205 · doi:10.1515/rose-2014-0006 |
| [11] | Kozachenko Yu.V., Veresh K.J. The heat equation with random initial conditions from Orlicz spaces. Theory Probab. Math. Statist. 2010, 80, 71-84. doi: 10.1090/S0094-9000-2010-00795-2 (translation of Teor. Imovir. Mat. Stat. 2009, 80, 63-75 (in Ukrainian)) · Zbl 1224.60113 · doi:10.1090/S0094-9000-2010-00795-2 |
| [12] | Kozachenko Yu.V., Veresh K.J. Boundary-value problem for a nonhomogeneous parabolic equation with Orlicz right side. Random Oper. Stoch. Equ. 2010, 18 (2), 97-119. doi: 10.1515/rose.2010.005 · Zbl 1226.35089 · doi:10.1515/rose.2010.005 |
| [13] | Slyvka-Tylyshchak A.I. Simulation of vibrations of a rectangular membrane with random initial conditions. Annales Mathematicae et Informaticae 2012, 39, 325-338. · Zbl 1265.60098 |
| [14] | Slyvka-Tylyshchak A.I. Justification of the Fourier method for equations of homogeneous string vibration with random initial conditions. Annales Univ. Sci. Budapest., Sect. Comp. 2012, 38, 211-232. · Zbl 1289.60092 |
| [15] | Slyvka G.I., Veresh K.J. Justifications of the Fourier method for hyperbolic equations with random initials conditions from Orlicz spaces. Bull. Uzhgorod Univ. Ser. Math. Inform. 2008, 16, 174-183. (in Ukrainian) · Zbl 1164.60374 |
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