Skip to main content
SpringerLink
Log in

On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

In this paper we give a positive answer to the question about the possibility of existence of several probability solutions to the Fokker–Planck–Kolmogorov equation for all initial conditions: we construct the first example of an equation with a unit diffusion matrix and a smooth drift coefficient for which the Cauchy problem with every probability initial condition has an infinite-dimensiona1 simplex of probability solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. N. Kolmogoroff, Math. Ann. 104, 415–458 (1931).

    Article  MathSciNet  Google Scholar 

  2. A. N. Kolmogoroff, Math. Ann. 104, 149–160 (1933).

    Article  MathSciNet  Google Scholar 

  3. V. I. Bogachev, N. V. Krylov, M. Röckner, and S. V. Shaposhnikov, Fokker–Planck–Kolmogorov Equ-ations (Am. Math. Soc., Providence, R.I., 2015).

    Book  Google Scholar 

  4. W. Feller, Math. Ann. 113 (1), 113–160 (1937).

    Article  MathSciNet  Google Scholar 

  5. K. Yosida, Ark. Mat. 1, 71–75 (1949).

    Article  MathSciNet  Google Scholar 

  6. E. Hille, J. Anal. Math. 3, 81–196 (1954).

    Article  Google Scholar 

  7. V. I. Bogachev, T. I. Krasovitskii, and S. V. Shaposhnikov, Dokl. Math. 102 (3), 464–467 (2020).

    Article  Google Scholar 

  8. V. I. Bogachev, T. I. Krasovitskii, and S. V. Shaposhnikov, Sb. Math. 212 (2021). https://doi.org/10.1070/SM9427

  9. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).

  10. T. I. Krasovitskii, Dokl. Math. 100 (1), 354–357 (2019).

    Article  Google Scholar 

  11. O. A. Oleinik and E. V. Radkevich, Second-Order Equations with Nonnegative Characteristic Form (Mosk. Gos. Univ. Moscow, 2010) [in Russian].

    MATH  Google Scholar 

  12. G. M. Fateeva, Math. USSR-Sb. 5 (4), 509–532 (1968).

    Article  Google Scholar 

Download references

Funding

This research was supported by the Russian Foundation for Basic Research (grant no. 20-01-00432) and the Moscow Center for Fundamental and Applied Mathematics. The third author acknowledges the support of the Simons Foundation; he is a winner of the “Young Russian Mathematics” contest and thanks its sponsors and jury.

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991, Moscow, Russia

    V. I. Bogachev, T. I. Krasovitskii & S. V. Shaposhnikov

  2. National Research University Higher School of Economics, 101000, Moscow, Russia

    V. I. Bogachev & S. V. Shaposhnikov

  3. St.-Tikhon’s Orthodox University, 115184, Moscow, Russia

    V. I. Bogachev

  4. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    V. I. Bogachev, T. I. Krasovitskii & S. V. Shaposhnikov

Authors
  1. V. I. Bogachev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. I. Krasovitskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. V. Shaposhnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Bogachev.

Additional information

The article was translated by the authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogachev, V.I., Krasovitskii, T.I. & Shaposhnikov, S.V. On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation. Dokl. Math. 103, 108–112 (2021). https://doi.org/10.1134/S1064562421030042

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562421030042

Keywords:

Search

Navigation