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

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Abstract

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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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 61–95, 2004

  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, 1997

  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, 2018

  5. Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Cinti, C., Pascucci, A., Polidoro, S.: Pointwise estimates for a class of non-homogeneous Kolmogorov equations. Math. Ann. 340, 237–264 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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, 1968

  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. https://doi.org/10.2422/2036-2145.201702_001

  12. Gualdani, M., Guillen, N.: On \(A_p\) weights and the Landau equation. Calc. Var. Partial Differ. Equ. 58, 17 (2019). https://doi.org/10.1007/s00526-018-1451-6

    Article  MATH  Google Scholar 

  13. Gutiérrez, C.E., Tournier, F.: Harnack Inequality for a Degenerate Elliptic Equation. Comm. Partial Differ. Equ. 36, 2103–2116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kogoj, A.E., Lanconelli, E., Tralli, G.: Wiener-Landis criterion for Kolmogorov-type operators. Discret. Contin. Dyn. Syst. 38, 2467–2485 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kolmogorov, A.N.: Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung). Ann. Math. 2(35), 116–117 (1934)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  18. Kuptsov, L.P.: Fundamental solutions of certain degenerate second-order parabolic equations. Math. Notes 31, 283–289 (1982)

    Article  MATH  Google Scholar 

  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. Lanconelli, A., Pascucci, A.: Nash Estimates and Upper Bounds for Non-homogeneous Kolmogorov equations. Potential Anal. 47, 461–483 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lanconelli, A., Pascucci, A., Polidoro, S.: Gaussian lower bounds for non-homogeneous Kolmogorov equations with measurable coefficients. Preprint: arXiv:1704.07307.pdf

  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, 2002

  23. Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Pol. Torino 52, 29–63 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Landis, E.M.: Harnack's inequality for second order elliptic equations of Cordes type. Dokl. Akad. Nauk SSSR 179, 1272–1275 (1968)

    MathSciNet  MATH  Google Scholar 

  25. Landis, E.M.: Second order equations of elliptic and parabolic type, vol. 171. American Mathematical Society, Translations of Mathematical Monographs, Providence, 1998

  26. Lieberman, G.M.: Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge (1996)

    Book  MATH  Google Scholar 

  27. Lions, P.L.: On Boltzmann and Landau equations. Philos. Trans. Roy. Soc. London Ser. A 346, 191–204, 1994

  28. Manfredini, M.: The Dirichlet problem for a class of ultraparabolic equations. Adv. Differ. Equ. 2, 831–866 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Pascucci, A., Polidoro, S.: The Moser's iterative method for a class of ultraparabolic equations. Commun. Contemp. Math. 6, 395–417 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Polidoro, S.: On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Le Matematiche (Catania) 49, 53–105 (1994)

    MathSciNet  MATH  Google Scholar 

  31. Polidoro, S.: A global lower bound for the fundamental solution of Kolmogorov-Fokker-Planck equations. Arch. Rational Mech. Anal. 137, 321–340 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Tralli, G.: A certain critical density property for invariant Harnack inequalities in H-type groups. J. Differ. Equ. 256, 461–474 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, W., Zhang, L.: The \(C^\alpha \) regularity of weak solutions of ultraparabolic equations. Discret. Contin. Dyn. Syst. 29, 1261–1275 (2011)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48823, USA

    Farhan Abedin

  2. Dipartimento d’Ingegneria Civile e Ambientale (DICEA), Università di Padova, Via Marzolo 9, 35131, Padova, Italy

    Giulio Tralli

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  1. Farhan Abedin
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  2. Giulio Tralli
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Correspondence to Giulio Tralli.

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Communicated by F. Lin

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Abedin, F., Tralli, G. Harnack inequality for a class of Kolmogorov–Fokker–Planck equations in non-divergence form. Arch Rational Mech Anal 233, 867–900 (2019). https://doi.org/10.1007/s00205-019-01370-z

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