Stationary solutions of a second-order differential equation with operator coefficients
Author:
M. F. Horodnii
Journal:
Theor. Probability and Math. Statist. 106 (2022), 177-181
MSC (2020):
Primary 60H99; Secondary 34G10
DOI:
https://doi.org/10.1090/tpms/1171
Published electronically:
May 16, 2022
MathSciNet review:
4438450
Full-text PDF
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Additional Information
Abstract: Necessary and sufficient conditions are given for the existence of a unique stationary solution to the second-order linear differential equation with bounded operator coefficients, perturbed by a stationary process. In the case when the corresponding “algebraic” operator equation has separated roots, the new representation of the stationary solution of the considered differential equation is obtained.
References
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- A. Ja. Dorogovcev, Some remarks on differential equations perturbed by periodic random processes, Ukrain. Mat. Ž. 14 (1962), 119–128 (Russian, with English summary). MR 0141850
- R. Z. Khas’minskii, Stability of Systems of Differential Equations under Random Perturbations of Their Parameters, “Nauka”, Moscow, Sijthoff and Noordhoff, Alphen aan Rijn, 1980.
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- A. G. Baskakov, T. K. Katsaran, and T. I. Smagina, Second-order linear differential equations in a Banach space and operator splitting, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (2017), 38–49 (Russian, with English and Russian summaries); English transl., Russian Math. (Iz. VUZ) 61 (2017), no. 10, 32–43. MR 3889179
References
- I. I. Vorovič, On stability of motion for random disturbances, Izv. Akad. Nauk SSSR. Ser. Mat. 20 (1956), 17–32. MR 0076999
- A. Ja. Dorogovcev, Some remarks on differential equations perturbed by periodic random processes, Ukrain. Mat. Ž. 14 (1962), 119–128. MR 0141850
- R. Z. Khas’minskii, Stability of Systems of Differential Equations under Random Perturbations of Their Parameters, “Nauka”, Moscow, Sijthoff and Noordhoff, Alphen aan Rijn, 1980.
- A. Ya. Dorogovtsev Periodic solutions of differential equations perturbed by stochastic processes, Ukrainian Math. J. 41, (1989), no. 12, 1412–1419. MR 1042961
- A. Ya. Dorogovtsev, Periodic and stationary regimes of infinite-dimensional deterministic and stochastic dynamical systems (Russian), “Vishcha Shkola”, Kiev, 1992. MR 1206004
- A. S. Markus, I. V. Mereutsa, On the Complete n-Tuple of Roots of the Operator Equation Corresponding to a Polynomial Operator Bundle, Math. USSR, Izv. 7, No. 5 (1973), 1105–1128. MR 719110
- A. G. Baskakov, T. K. Katsaran, T. I. Smagina, Second-order linear differential equations in a Banach space and splitting operators, Russian Math., 61 (2017), no. 10, 32–43. MR 3889179
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Additional Information
M. F. Horodnii
Affiliation:
Taras Shevchenko National University of Kyiv, 64, Volodymyrs’ka St., 01601 Kyiv, Ukraine
Email:
horodnii@gmail.com
Keywords:
Banach space,
second-order differential equation,
stationary solution,
“algebraic” operator equation,
separated roots
Received by editor(s):
January 5, 2022
Accepted for publication:
January 24, 2022
Published electronically:
May 16, 2022
Additional Notes:
This work has been supported by Ministry of Education and Science of Ukraine: Grant of the Ministry of Education and Science of Ukraine for perspective development of a scientific direction “Mathematical sciences and natural sciences” at Taras Shevchenko National University of Kyiv.
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv
