Harnack inequality for a class of Kolmogorov–Fokker–Planck equations in non-divergence form

  • Farhan Abedin
  • Giulio TralliEmail author


We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient matrix is assumed either to satisfy a Cordes–Landis condition on the eigenvalues, or to admit a uniform modulus of continuity.


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F.A. wishes to thank Prof. Brian Rider for providing financial support through his NSF Grant DMS–1406107. G.T. has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors would like to thank Prof. Luis Silvestre for suggesting this problem at the 2017 Chicago Summer School in Analysis, and the anonymous referee for their valuable comments on the manuscript.

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Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Abedin, F., Gutiérrez, C.E., Tralli, G.: Harnack's inequality for a class of non-divergent equations in the Heisenberg group. Comm. Partial Differ. Equ. 42, 1644–1658 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 61–95, 2004Google Scholar
  3. 3.
    Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance theory. In: `Numerical methods in finance'. Publications of the Newton Institute Cambridge University Press, Cambridge, 13, pp. 1–21, 1997Google Scholar
  4. 4.
    Cameron, S., Silvestre, L., Snelson, S.: Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 625–642, 2018Google Scholar
  5. 5.
    Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cinti, C., Pascucci, A., Polidoro, S.: Pointwise estimates for a class of non-homogeneous Kolmogorov equations. Math. Ann. 340, 237–264 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Delarue, F., Menozzi, S.: Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259, 1577–1630 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Di Francesco, M., Polidoro, S.: Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Differ. Equ. 11, 1261–1320 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Garofalo, N., Lanconelli, E.: Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type. Trans. Amer. Math. Soc. 321, 775–792 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Glagoleva, R. Ya.: A priori estimate of the Hölder norm and the Harnack inequality for the solution of a second order linear parabolic equation with discontinuous coefficients. Mat. Sb. (N.S.) 76, 167–185, 1968Google Scholar
  11. 11.
    Golse, F., Imbert, C., Mouhot, C., Vasseur, A.F.: Harnack inequality for kinetic Fokker–Planck equations with rough coefficients and application to the Landau equation. Ann. Sc. Norm. Super. Pisa Cl. Sci.
  12. 12.
    Gualdani, M., Guillen, N.: On \(A_p\) weights and the Landau equation. Calc. Var. Partial Differ. Equ. 58, 17 (2019). CrossRefzbMATHGoogle Scholar
  13. 13.
    Gutiérrez, C.E., Tournier, F.: Harnack Inequality for a Degenerate Elliptic Equation. Comm. Partial Differ. Equ. 36, 2103–2116 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kogoj, A.E., Lanconelli, E., Tralli, G.: Wiener-Landis criterion for Kolmogorov-type operators. Discret. Contin. Dyn. Syst. 38, 2467–2485 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kolmogorov, A.N.: Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung). Ann. Math. 2(35), 116–117 (1934)CrossRefzbMATHGoogle Scholar
  17. 17.
    Krylov, N.V., Safonov, M.V.: A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44, 161–175 (1980)MathSciNetGoogle Scholar
  18. 18.
    Kuptsov, L.P.: Fundamental solutions of certain degenerate second-order parabolic equations. Math. Notes 31, 283–289 (1982)CrossRefzbMATHGoogle Scholar
  19. 19.
    Imbert, C., Silvestre, L.: Weak Harnack inequality for the Boltzmann equation without cut-off. J. Eur. Math. Soc. (JEMS). Preprint: arXiv:1608.07571.pdf
  20. 20.
    Lanconelli, A., Pascucci, A.: Nash Estimates and Upper Bounds for Non-homogeneous Kolmogorov equations. Potential Anal. 47, 461–483 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lanconelli, A., Pascucci, A., Polidoro, S.: Gaussian lower bounds for non-homogeneous Kolmogorov equations with measurable coefficients. Preprint: arXiv:1704.07307.pdf
  22. 22.
    Lanconelli, E., Pascucci, A., Polidoro, S.: Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. In: ``Nonlinear problems in mathematical physics and related topics,II'', Interactive Mathematics Series (N. Y.) 2, pp. 243–265, 2002Google Scholar
  23. 23.
    Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Pol. Torino 52, 29–63 (1994)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Landis, E.M.: Harnack's inequality for second order elliptic equations of Cordes type. Dokl. Akad. Nauk SSSR 179, 1272–1275 (1968)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Landis, E.M.: Second order equations of elliptic and parabolic type, vol. 171. American Mathematical Society, Translations of Mathematical Monographs, Providence, 1998Google Scholar
  26. 26.
    Lieberman, G.M.: Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge (1996)CrossRefzbMATHGoogle Scholar
  27. 27.
    Lions, P.L.: On Boltzmann and Landau equations. Philos. Trans. Roy. Soc. London Ser. A 346, 191–204, 1994Google Scholar
  28. 28.
    Manfredini, M.: The Dirichlet problem for a class of ultraparabolic equations. Adv. Differ. Equ. 2, 831–866 (1997)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Pascucci, A., Polidoro, S.: The Moser's iterative method for a class of ultraparabolic equations. Commun. Contemp. Math. 6, 395–417 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Polidoro, S.: On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Le Matematiche (Catania) 49, 53–105 (1994)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Polidoro, S.: A global lower bound for the fundamental solution of Kolmogorov-Fokker-Planck equations. Arch. Rational Mech. Anal. 137, 321–340 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tralli, G.: A certain critical density property for invariant Harnack inequalities in H-type groups. J. Differ. Equ. 256, 461–474 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, W., Zhang, L.: The \(C^\alpha \) regularity of weak solutions of ultraparabolic equations. Discret. Contin. Dyn. Syst. 29, 1261–1275 (2011)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Dipartimento d’Ingegneria Civile e Ambientale (DICEA)Università di PadovaPadovaItaly

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