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Degenerate Elliptic Equations and Nonuniqueness of Solutions to the Kolmogorov Equation

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Abstract

In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker–Planck–Kolmogorov equation to a degenerate elliptic equation on a bounded domain.

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REFERENCES

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

    Article  MathSciNet  Google Scholar 

  2. V. I. Bogachev, M. Röckner, and W. Stannat, Sb. Math. 193 (7), 945–976 (2002).

    Article  MathSciNet  Google Scholar 

  3. S. V. Shaposhnikov, J. Funct. Anal. 254 (10), 2690–2705 (2008).

    Article  MathSciNet  Google Scholar 

  4. M. A. Pozio, F. Punzo, and A. Tesei, J. Math. Pures Appl. 90, 353–386 (2008).

  5. V. I. Bogachev, T. I. Krasovitskii, and S. V. Shaposhnikov, Dokl. Math. 98 (2), 475–479 (2018).

    Article  Google Scholar 

  6. R. Z. Khasminskii, Stochastic Stability of Differential Equations (Sijthoff and Noordhoff, 1980).

    Book  Google Scholar 

  7. V. V. Zhikov, Funct. Anal. Appl. 38 (3), 173–183 (2004).

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

    Book  Google Scholar 

  9. O. A. Oleinik and E. V. Radkevic, Second Order Equations with Nonnegative Characteristic Form (Plenum, New York, 1973).

    Book  Google Scholar 

  10. O. A. Oleinik, Mat. Sb. 69 (111), 111–140 (1966).

    MathSciNet  Google Scholar 

  11. A. Friedman and A. Pinsky, Trans. Am. Math. Soc. 186, 359–383 (1973).

    Article  Google Scholar 

  12. N. S. Nadirashvili, Ann. Sc. Norm. Sup. Pisa Cl. Sci 24 (4), 537–550 (1997).

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Funding

This paper has been supported by the Russian Foundation for Basic Research (grant 18-31-20008) and the DFG (grant RO 1195/12-1).

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Authors and Affiliations

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

    T. I. Krasovitskii

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  1. T. I. Krasovitskii
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Correspondence to T. I. Krasovitskii.

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Krasovitskii, T.I. Degenerate Elliptic Equations and Nonuniqueness of Solutions to the Kolmogorov Equation. Dokl. Math. 100, 354–357 (2019). https://doi.org/10.1134/S1064562419040112

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  • DOI: https://doi.org/10.1134/S1064562419040112

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