Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity

Abstract

We prove a formula representing solutions to parabolic Fokker–Planck–Kolmogorov equations with coefficients of low regularity. This formula is applied for proving the continuity of solution densities under broad assumptions and obtaining upper bounds for them. In the case of diffusion coefficients of class VMO\(_x\), we show that the solution density is locally integrable to any power.

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References

  1. 1.

    Bally, V., Caramellino, L.: Convergence and regularity of probability laws by using an interpolation method. Ann. Probab. 45:2 (2017), 1110–1159.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bauman, P.: Equivalence of the Green’s functions for diffusion operators in \({\rm R}^n\): a counterexample. Proc. Amer. Math. Soc. 91 (1984), 64–68.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bogachev, V.I.: Measure theory. V. 1, 2, Springer, Berlin, 2007.

    Book  Google Scholar 

  4. 4.

    Bogachev, V.I., Krylov, N.V., Röckner, M.: On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm. Partial Differ. Equ. 26 (2001), 2037–2080.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bogachev, V.I., Krylov, N.V., Röckner, M.: Elliptic and parabolic equations for measures. Russian Math. Surveys 64:6 (2009), 973–1078.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker–Planck–Kolmogorov equations. Amer. Math. Soc., Providence, Rhode Island, 2015.

    Book  Google Scholar 

  7. 7.

    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: Global regularity and bounds for solutions of parabolic equations for probability measures. Teor. Verojatn. Primen. 50:4 (2005), 652–674 (in Russian); English transl.: Theory Probab. Appl. 50:4 (2006), 561–581.

    Article  Google Scholar 

  8. 8.

    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: Estimates of densities of stationary distributions and transition probabilities of diffusion processes. Teor. Verojatn. i Primen. 52:2 (2007), 240–270 (in Russian); English transl.: Theory Probab. Appl. 52:2 (2008), 209–236.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Bogachev, V.I., Shaposhnikov, S.V.: Integrability and continuity of solutions to double divergence form equations. Annali di Matematica Pura ed Applicata 196:5 (2017), 1609–1635.

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fabes, E.B., Kenig, C.E.: Examples of singular parabolic measures and singular transition probability densities. Duke Math. J. 48 (1981), 845–856.

    MathSciNet  Article  Google Scholar 

  11. 11.

    Fournier, N., Printems, J.: Absolute continuity for some one-dimensional processes. Bernoulli 16:2 (2010), 343–360.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, New Jersey, 1964.

    MATH  Google Scholar 

  13. 13.

    Krylov, N.V.: Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces. J. Funct. Anal. 183:1 (2001), 1–41.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Krylov, N.V.: Parabolic and elliptic equations with VMO coefficients. Comm. Partial Differ. Equ. 32:1-3 (2007), 453–475.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Krylov, N.V.: Lectures on elliptic and parabolic equations in Sobolev spaces. Amer. Math. Soc., Providence, Rhode Island, 2008.

    Book  Google Scholar 

  16. 16.

    Lieberman, G.: A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with \(VMO\) coefficients. J. Funct. Anal. 201 (2003), 457–479.

    MathSciNet  Article  Google Scholar 

  17. 17.

    Metafune, G., Pallara, D., Rhandi, A.: Global properties of invariant measures. J. Funct. Anal. 223 (2005), 396–424.

    MathSciNet  Article  Google Scholar 

  18. 18.

    Metafune, G., Pallara, D., Rhandi, A.: Global properties of transition probabilities of singular diffusions. Theory Probab. Appl. 54:1 (2009), 116–148.

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Sjögren, P.: On the adjoint of an elliptic linear differential operator and its potential theory. Ark. Mat. 11 (1973), 153–165.

    MathSciNet  Article  Google Scholar 

  20. 20.

    Sjögren, P.: Harmonic spaces associated with adjoints of linear elliptic operators. Annal. Inst. Fourier 25:3-4 (1975), 509–518.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This research was supported by the RFBR Grants 17-01-00662 and 18-31-20008 and the DFG through the project RO 1195/12-1 and the CRC 1283 at Bielefeld University. We are very grateful to the anonymous referee for important corrections and useful remarks taken into account in the final version.

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Correspondence to Vladimir I. Bogachev.

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Bogachev, V.I., Shaposhnikov, S.V. Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity. J. Evol. Equ. 20, 355–374 (2020). https://doi.org/10.1007/s00028-019-00532-6

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Keywords

  • Fokker–Planck–Kolmogorov equation
  • Double divergence form equation
  • Representation of solutions

Mathematics Subject Classification

  • Primary 35J15
  • Secondary 35B65