# Properties of solutions to linear KdV equations with φ-sub-Gaussian initial conditions

## DOI:

https://doi.org/10.17721/1812-5409.2022/2.1## Keywords:

φ-sub-Gaussian processes, Airy equation, random initial condition, distribution of supremum## Abstract

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded domains. Also, there are presented some examples to illustrate the results of the study.

* Pages of the article in the issue*: 11 - 19

** Language of the article**: English

## References

BEGHIN, L., KNOPOVA, V., LEONENKO, N., ORSINGHER, E. (2000) Gaussian Limiting Behavior of the Rescaled Solution to the Linear Korteweg de Vries Equation with Random Initial Conditions. J. Stat. Phys. Vol. 99, Iss. 3/4, p. 769–781.

BEGHIN, L., KOZACHENKO, YU., ORSINGHER, E., SAKHNO, L. (2007) On the Solutions of Linear Odd-Order Heat-Type Equations with Random Initial Conditions. J. Stat. Phys. Vol. 127, Iss. 4, p. 721–739.

GIULIANO ANTONINI R., KOZACHENKO YU., NIKITINA T. (2003) Space of φ-sub-Gaussian random variables. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5). Vol. 27. P. 92–124.

BULDYGIN, V. V., KOZACHENKO, YU. V. (2000) Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence, RI. 257 p.

HOPKALO, O., SAKHNO, L. (2021) Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes. Modern Stoch. Theory Appl. Vol. 8, Iss. 1. P. 41–62.

HOPKALO, O. M., SAKHNO, L. M., VASYLYK, O. I. (2020) Properties of φ-sub-Gaussian processes related to the heat equation with random initial conditions. Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics & Mathematics. Vol. 1-2. P. 17-24.

KOZACHENKO, YU., V., LEONENKO, G. M. (2002) Large deviations type inequality for the supremum of the heat random field. Methods Func. Anal. Topol. Vol. 8, No. 3. P. 46–49.

KOZACHENKO, YU., V., LEONENKO, G. M. (2006) Extremal behavior of the heat random field. Extremes. Vol. 8. P. 191–205.

KOZACHENKO, YU. V., KOVAL’CHUK, YU. A. (1998) Boundary value problems with random initial conditions and series of functions of Subφ(Ω). Ukr. Math. J. 50, p. 572–585.

KOZACHENKO, YU., ORSINGHER, E., SAKHNO, L., VASYLYK, O. (2018) Estimates for functional of solution to higher-order heat-type equation with random initial condition. J. Stat. Phys.. Vol. 72. P. 1641–1662.

KOZACHENKO, YU., ORSINGHER, E., SAKHNO, L., VASYLYK, O. (2020) Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions. Modern Stoch. Theory Appl., Vol. 7, Iss. 1, p. 79–96.

KOZACHENKO, YU. V., OSTROVSKY, E. I. (1985) Banach spaces of random variables of sub-Gaussian type. Theor. Probab. Math. Stat. No. 32, p.42–53.

KOZACHENKO, YU. V., SLIVKA, G.I. (2004). Justification of the Fourier method for hyperbolic equations with random initial conditions. Theor. Probab. Math. Stat. Vol. 69. P. 67-83.

KOZACHENKO, YU. V., SLIVKA-TYLYSHCHAK, A. I. (2014) On the increase rate of random fields from space Subφ(Ω) on unbounded domains. Stat. Optim. Inf. Comput. 2, No. 2. P. 79–92.

KOZACHENKO, YU., SOTTINEN T., VASYLYK, O. (2011) Lipschitz conditions for Subφ(Ω)-processes and applications to weakly self-similar processes with stationary increments. Theor. Probab. Math. Stat. Vol. 82. P. 57–73.

KRASNOSEL’SKII, M. A., RUTICKII, YA. B. (1961) Convex Functions and Orlicz Spaces. P.Noordhoff Ltd, Groningen. 249p.

SAKHNO, L. M., VASYLYK, O. I. (2021) Investigation of solutions to higher-order dispersive equations with φ-sub-Gaussian initial conditions. Bulletin of Taras Shevchenko National Universiry of Kyiv. Series: Physics & Mathematics. Vol.2. P. 78 - 84. 18. TAO, T. (2006) Nonlinear Dispersive Equations: Local and Global Analysis. CBMS Regional Conf. Series in Math. V.106, AMS. 373p.

VASYLYK, O. I., KOZACHENKO, YU. V., YAMNENKO, R. E. (2008) φ-sub-Gaussian random process. Kyiv: Vydavnycho-Poligrafichnyi Tsentr “Kyivskyi Universytet”, 231 p. (In Ukrainian)

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (2), 11–19. https://doi.org/10.17721/1812-5409.2022/2.1

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