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Doklady Mathematics

, Volume 98, Issue 3, pp 559–563 | Cite as

Estimates for Solutions to Fokker–Planck–Kolmogorov Equations with Integrable Drifts

  • V. I. Bogachev
  • A. V. Shaposhnikov
  • S. V. Shaposhnikov
Mathematics
  • 3 Downloads

Abstract

The result of this paper states that every probability measure satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a -integrable perturbation of the drift term–x of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure γ and \(f = \frac{{d\mu }}{{d\gamma }}\) for the density the integral of
$$f\left| {\log } \right|{\left( {f + 1} \right)^\alpha }$$
with respect to γ is estimated via \({\left\| v \right\|_{{L^1}\left( \mu \right)}}\) for all α < \(\frac{1}{4}\). This shows that stationary measures of infinite-dimensional diffusions whose drifts are integrable perturbations of–are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds.

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References

  1. 1.
    V. I. Bogachev, N. V. Krylov, and M. Röckner, Commun. Partial Differ. Equations 26, 2037–2080 (2001).CrossRefGoogle Scholar
  2. 2.
    V. I. Bogachev, N. V. Krylov, M. Röckner, and S. V. Shaposhnikov, Fokker–Planck–Kolmogorov Equations (Am. Math. Soc., Providence, R.I., 2015).CrossRefzbMATHGoogle Scholar
  3. 3.
    V. I. Bogachev and M. Röckner, J. Funct. Anal. 133, 168–223 (1995).MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. I. Bogachev, S. N. Popova, and S. V. Shaposhnikov, Dokl. Math. 98 (2) (2018). arXiv: 1803.04568Google Scholar
  5. 5.
    M. Ledoux, Lect. Notes Math. 1648, 165–294 (1996).CrossRefGoogle Scholar
  6. 6.
    M. Fukushima and M. Hino, J. Funct. Anal. 183 (1), 245–268 (2001).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. I. Bogachev, Gaussian Measures (Am. Math. Soc., Providence, R.I., 1998).CrossRefzbMATHGoogle Scholar
  8. 8.
    V. I. Bogachev, Russ. Math. Surv. 73 (2), 191–260 (2018).CrossRefGoogle Scholar
  9. 9.
    V. I. Bogachev and A. V. Kolesnikov, Russ. Math. Surv. 67 (5), 785–890 (2012).CrossRefGoogle Scholar
  10. 10.
    F.-Y. Wang, Probab. Theory Relat. Fields 109, 417–424 (1997).CrossRefGoogle Scholar
  11. 11.
    F.-Y. Wang, Analysis for Diffusion Processes on Riemannian Manifolds (World Sci., Singapore, 2014).zbMATHGoogle Scholar
  12. 12.
    V. I. Bogachev, E. D. Kosov, and G. I. Zelenov, Trans. Am. Math. Soc. 370 (6), 4401–4432 (2018).CrossRefGoogle Scholar
  13. 13.
    L. Ambrosio and A. Figalli, Ann. Fac. Sci. Toulouse Math. 20 (2), 407–438 (2011).MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Röckner and F.-Y. Wang, Infinite Dimen. Anal. Quantum Probab. Relat. Topics 13 (1), 27–37 (2010).CrossRefGoogle Scholar
  15. 15.
    V. I. Bogachev, M. Röckner, and F.-Y. Wang, J. Math. Pures Appl. 80, 177–221 (2001).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • V. I. Bogachev
    • 1
    • 2
    • 3
  • A. V. Shaposhnikov
    • 1
  • S. V. Shaposhnikov
    • 1
    • 2
    • 3
  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.St. Tikhon’s Orthodox UniversityMoscowRussia

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