On the third initial-boundary value problem for some class of pseudo-differential equations related to a symmetric \(\alpha \)-stable process

  • M. M. OsypchukEmail author
  • M. I. Portenko


A fundamental solution of the so-called third initial-boundary value problem for one class of pseudo-differential equations is constructed. Those equations are related to a symmetric \(\alpha \)-stable stochastic process and our constructions are inspired by some probabilistic ideas. However, we expound our results in a way completely independent of any probabilistic notion. Only the last section of the paper is based on the notion of a stochastic process and also a pseudo-process and it gives some interpretation of our results in terms of stochastic analysis.


Pseudo-differential equations Initial-boundary value problem Fundamental solution Single-layer potential \(\alpha \)-Stable stochastic process 

Mathematics Subject Classification

Primary 35S11 Secondary 60G52 


  1. 1.
    Beghin, L., Orsinger, E.: The distribution of the local time for “pseudoprocesses” and its connection with fractional diffusion equations. Stoch. Process. Appl. 115, 1017–1040 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dynkin, E.B.: Markov Processes. Fizmatgiz, Moscow (1963); English transl., vols. I, II. Academic Press, New-York and Springer, Berlin (1965)Google Scholar
  3. 3.
    Eidelman, S.D., Ivasyshen, S.D., Kochubei, A.N.: Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type. Operator Theory: Advances and Applications, vol. 152. Birkhäuser, Basel (2004)zbMATHGoogle Scholar
  4. 4.
    Fridman, A.: Partial Differential Equations of Parabolic Type. Prentige-Hall Inc., Englewood Cliffs (1964)Google Scholar
  5. 5.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)CrossRefGoogle Scholar
  6. 6.
    Löbus, J.-U., Portenko, M.I.: On one class of perturbations of the generator of a stable process. Theory Probab. Math. Stat. 52, 102–111 (1995)zbMATHGoogle Scholar
  7. 7.
    Osypchuk, M.M., Portenko, M.I.: One type of singular perturbations of a multidimensional stable process. Theory Stoch. Process. 19(2), 42–51 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Osypchuk, M.M., Portenko, M.I.: On simple-layer potentials for one class of pseudo-differential equations. Ukr. Math. J. 67(11), 1704–1720 (2016)CrossRefGoogle Scholar
  9. 9.
    Podolynny, S.I., Portenko, N.I.: On multidimentional stable processes with locally unbounded drift. Random Oper. Stoch Equ. 3(2), 113–124 (1995)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Statistics and High MathematicsVasyl Stefanyk Precarpathian National UniversityIvano-FrankivskUkraine
  2. 2.Department of Theory of Random ProcessesInstitute of Mathematics of the National Academy of Sciences of UkraineKyivUkraine

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