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

Authors

  • O. M. Hopkalo Taras Shevchenko National University of Kyiv
  • L. M. Sakhno Taras Shevchenko National University of Kyiv
  • O. I. Vasylyk Taras Shevchenko National University of Kyiv https://orcid.org/0000-0002-0880-3751

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.

No 1-2 (2020)

Downloads

How to Cite

Hopkalo, O. M., Sakhno, L. M., & Vasylyk, O. I. (2020). Properties of $\varphi$-sub-Gaussian stochastic processes related to the heat equation with random initial conditions. 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

Issue

No. 1-2 (2020)

Section

Algebra, Geometry and Probability Theory

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

Most read articles by the same author(s)