Cybernetics and Systems Analysis

, Volume 54, Issue 2, pp 278–283 | Cite as

Stochastic m-Point Cauchy Problem for Parabolic Equation with Semi-Wiener Perturbations

  • G. M. Perun
  • V. K. Yasinsky


The authors consider the problem of the existence, with probability 1, of Green’s function of a stochastic m-point Cauchy problem for a higher-order parabolic equation with white noise perturbations taken with negative values only. Estimates are obtained in spaces of functions whose norm contains the mathematical expectation.


m-point Cauchy problem existence with probability 1 Ito’s formula Green’s function parabolicity condition mathematical expectation 


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  1. 1.
    M. I. Matiichuk, Parabolic Singular Boundary-Value Problems [in Ukrainian], Inst. Mathem. NAS of Ukraine, Kyiv (1999).Google Scholar
  2. 2.
    V. V. Gorodetsky and D. I. Spizhavka, “Multipoint problem for evolutionary equations with pseudo-Bessel operators,” Dopov. Nac. Akad. Nauk Ukr., No. 12, 7–12 (2009).Google Scholar
  3. 3.
    I. I. Gikhman, “Boundary-value problem for a stochastic equation of parabolic type,” Ukr. Math. J., Vol. 31, No. 5, 383–387 (1979).Google Scholar
  4. 4.
    I. I. Gikhman, “A mixed problem for a stochastic differential equation of parabolic type,” Ukr. Math. J., Vol. 32, No. 3, 243–246 (1980).Google Scholar
  5. 5.
    V. K. Yasinsky and N. P. Bodryk, “Investigating properties of strong solution to the Caushy problem for linear stochastic partial differential-difference equation with Markov parameters," Naukovyi Visnyk Chernivets’kogo Univ. im. Yu. Fed’kovicha, Ser. Matem., Vol. 1, No. 1–2, 158–167 (2011).Google Scholar
  6. 6.
    G. M. Perun, “Problem with pulse action for a linear stochastic parabolic equation of higher order," Ukr. Math. J., Vol. 60, No. 10, 1660–1665 (2008).Google Scholar
  7. 7.
    I. I. Gikhman and A. V. Skorokhod, An Introduction to the Theory of Random Processes [in Russian], Nauka, Moscow (1977).Google Scholar
  8. 8.
    S. D. Eidelman, Parabolic Systems [in Russian], Nauka, Moscow (1964).Google Scholar
  9. 9.
    M. L. Sverdan, E. F. Tsarkov, and V. K. Yasinsky, Theory of Random Processes [in Ukrainian], Zoloti Lytavry, Chernivtsi (2008).Google Scholar
  10. 10.
    V. K. Yasinsky, Mathematical Theory of Random Processes [in Ukrainian], Rodovid, Chernivtsi (2014).Google Scholar
  11. 11.
    A. V. Dorogovtsev, Stochastic Analysis and Random Transformation in Hilbert Space [in Russian], Naukova Dumka, Kyiv (1992).zbMATHGoogle Scholar
  12. 12.
    A. Yu. Shevlyakov, “Ito’s formula for extended stochastic integral,” Probability Theory and Math. Statistics, Issue 22, 146–157 (1982).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Yu. Fedkovych Chernivtsi National UniversityChernivtsiUkraine

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