# Properties of $\varphi$-sub-Gaussian stochastic processes related to the heat equation with random initial conditions

## DOI:

https://doi.org/10.17721/1812-5409.2020/1-2.2## Abstract

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in $L_2(\Omega)$ sense) of the heat equation with random initial conditions given by $\varphi$-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded and unbounded domains.

* Key words*: ?-sub-Gaussian processes, heat equation, random initial condition, distribution of supremum.

* Pages of the article in the issue*: 17 - 24

** Language of the article**: English

## References

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, Issue 4, p. 721-739.

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

KOZACHENKO, YU. V., LEONENKO, G.M. (2002) Large deviations type inequality for the supremum of the heat random field. Methods Func. Anal. Topol. 8 (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 $Sub_varphi(Omega)$. 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. 72, p. 1641--1662.

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. 69, p. 67-83.

KOZACHENKO, YU. V., SLYVKA-TYLYSHCHAK, A.I. (2014) On the increase rate of random fields from space $Sub_varphi(Omega)$ on unbounded domains. textit{Stat. Optim. Inf. Comput. 2, No. 2, p. 79-92.

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

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (1-2), 17–24. https://doi.org/10.17721/1812-5409.2020/1-2.2

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