Siberian Mathematical Journal

, Volume 58, Issue 6, pp 1090–1103 | Cite as

On the Inhomogeneous Conservative Wiener–Hopf Equation



We prove the existence of a solution to the inhomogeneous Wiener–Hopf equation whose kernel is a probability distribution generating a random walk drifting to −∞. Asymptotic properties of a solution are found depending on the corresponding properties of the free term and the kernel of the equation.


integral equation inhomogeneous equation inhomogeneous generalized Wiener–Hopf equation probability distribution drift to minus infinity asymptotic behavior 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Feller W., An Introduction to Probability Theory and Its Applications. Vol. 2, John Wiley and Sons, Inc., New York etc. (1966).MATHGoogle Scholar
  2. 2.
    Neveu J., Mathematical Foundations of Probability Theory [Russian translation], Mir, Moscow (1969).Google Scholar
  3. 3.
    Norkin V. I., “Solving the Wiener–Hopf equation with a probabilistic kernel,” Cybernetics and Systems Analysis, vol. 42, no. 2, 195–201 (2006).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dmitriev V. I., “The Wiener–Hopf equation,” in: Mathematical Encyclopedia [Russian], Sovetskaya Èntsiklopediya, Moscow, 1977, vol. 1, 967–968.Google Scholar
  5. 5.
    Stone C., “On absolutely continuous components and renewal theory,” Ann. Math. Stat., vol. 37, 271–275 (1966).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Alsmeyer G., Erneuerungstheorie, B. G. Teubner, Stuttgart (1991).CrossRefMATHGoogle Scholar
  7. 7.
    Arjas E., Nummelin E., and Tweedie R. L., “Uniform limit theorems for non-singular renewal and Markov renewal processes,” J. Appl. Probab., vol. 15, 112–125 (1978).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Sgibnev M. S., “Equivalence of two conditions on singular components,” Stat. Probab. Lett., vol. 40, 127–131 (1998).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sgibnev M. S., “Stone’s decomposition of the renewal measure via Banach-algebraic techniques,” Proc. Amer. Math. Soc., vol. 130, 2425–2430 (2002).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hille E. and Phillips R. S., Functional Analysis and Semigroups, Amer. Math. Soc., Providence (1957).MATHGoogle Scholar
  11. 11.
    Sgibnev M. S., “Submultiplicative moments of the supremum of a random walk with negative drift,” Stat. Probab. Lett., vol. 32, 377–383 (1997).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Sgibnev M. S., “Semimultiplicative moments of factors in Wiener-Hopf matrix factorization,” Sb. Math., vol. 199, no. 2, 277–290 (2008).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gelfand I. M., Raĭkov D. A., and Shilov G. E., Commutative Normed Rings, Chelsea Publishing Company, New York (1964).Google Scholar
  14. 14.
    Rogozin B. A. and Sgibnev M. S., “Banach algebras of measures on the line,” Sib. Math. J., vol. 21, no. 2, 265–273 (1980).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fikhtengol’ts G. M., A Course of Differential and Integral Calculus. Vol. 1 [Russian], Nauka, Moscow (1966).Google Scholar
  16. 16.
    Sgibnev M. S., “An asymptotic expansion for the distribution of the supremum of a random walk,” Studia Math., vol. 140, 41–55 (2000).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations