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From Döblin to Aleksandrov

Primarily AI-generated textHuman understanding: all partsmath.AP — Analysis of PDEsmath.PR — Probability

Contributed by Amélie Loher ↗, Connor Mooney, Clément Mouhot

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We prove Aleksandrov maximum principles for linear second-order kinetic equations in non-divergence form with measurable, uniformly elliptic diffusion coefficients. For coefficients A(t,v)A(t,v) we obtain every source exponent p>2d+1p>2d+1. For autonomous coefficients a(x,v)a(x,v) in one space dimension, we obtain every p>p∗(Λλ)p>p_*(\frac\Lambda\lambda), where p∗(Λλ)<4p_*(\frac\Lambda\lambda)<4 depends only on the ellipticity ratio. The first step in both proofs is Döblin--Fourier cancellation: two positive parts of the kernel have nearly opposite phases, and a common lower bound for their velocity marginals allows us to cancel fixed portions of them. For the autonomous problem, this argument applies on velocity intervals away from zero. A preliminary Aleksandrov estimate gives Hölder continuity and then a Harnack comparison. A homogeneous Bellman barrier controls concentration in position and velocity; together with Harnack, it makes the local density estimates summable across zero velocity, including at p=4p=4. We give the full proofs and the passage from the maximum principle to local Hölder continuity.

Provenance statement

This paper is accompanying a human-written paper, "Döblin–Fourier cancellation and kinetic Aleksandrov estimates", arXiv:2610.03083. The result of both these papers were generated through human notes collected over 2 years, extensive discussions with the AI over 5 months. The results have been further developed with the assistance of AI and digested humanly in "Döblin–Fourier cancellation and kinetic Aleksandrov estimates". The paper here is the companion paper with all the technicalities and details for a complete proof. This paper is lean-formalised https://github.com/amelieloher/hypoelliptic-aleksandrov.

Formalizations

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References

  1. F. Abedin and G. Tralli, Harnack inequality for a class of Kolmogorov--Fokker--Planck equations in non-divergence form, Arch. Ration. Mech. Anal. 233 (2019), no. 2, 867--900. doi:10.1007/s00205-019-01370-z (https://doi.org/10.1007/s00205-019-01370-z)DOI
  2. A. D. Aleksandrov, Certain estimates for the Dirichlet problem, Dokl. Akad. Nauk SSSR 134 (1960), 1001--1004 (Russian); English transl.\ Soviet Math. Dokl. 1 (1960), 1151--1154.
  3. I. Ya. Bakelman, Theory of quasilinear elliptic equations, Sibirsk. Mat. Zh. 2 (1961), 179--186 (Russian).
  4. NIST Digital Library of Mathematical Functions, Chapter 13: Confluent hypergeometric functions. https://dlmf.nist.gov/13.link
  5. W. Doeblin, Éléments d'une théorie générale des chaînes simples constantes de Markoff , Ann. Sci. École Norm. Sup. (3) 57 (1940), 61--111.
  6. H. Dong and T. Yastrzhembskiy, Global LpL_p estimates for kinetic Kolmogorov--Fokker--Planck equations in nondivergence form, Arch. Ration. Mech. Anal. 245 (2022), no. 1, 501--564. doi:10.1007/s00205-022-01786-0 (https://doi.org/10.1007/s00205-022-01786-0)DOI
  7. E. Ferretti and M. V. Safonov, Growth theorems and Harnack inequality for second order parabolic equations, in: Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000), Contemp. Math., vol. 277, American Mathematical Society, Providence, RI, 2001, pp. 87--112. doi:10.1090/conm/277/04540 (https://doi.org/10.1090/conm/277/04540)DOI
  8. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order , 2nd ed., Grundlehren der mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1983; reprinted in Classics in Mathematics, Springer, Berlin, 2001. doi:10.1007/978-3-642-61798-0 (https://doi.org/10.1007/978-3-642-61798-0)DOI
  9. F. Golse, C. Imbert, C. Mouhot and A. F. Vasseur, Harnack inequality for kinetic Fokker--Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 253--295. doi:10.2422/2036-2145.201702_001 (https://doi.org/10.2422/2036-2145.201702_001)DOI
  10. J. Guerand and C. Imbert, Log-transform and the weak Harnack inequality for kinetic Fokker--Planck equations, J. Inst. Math. Jussieu 22 (2023), no. 6, 2749--2774. doi:10.1017/S1474748022000160 (https://doi.org/10.1017/S1474748022000160)DOI
Show all 24 referencesShow fewer references
  1. J. Guerand and C. Mouhot, Quantitative De Giorgi methods in kinetic theory, J. Éc. polytech. Math. 9 (2022), 1159--1181. doi:10.5802/jep.203 (https://doi.org/10.5802/jep.203)DOI
  2. Z. Hao and X. Zhang, Harnack estimates for nondivergence kinetic equations, preprint (2026), arXiv:2609.15534 (https://arxiv.org/abs/2609.15534).arXiv
  3. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147--171. doi:10.1007/BF02392081 (https://doi.org/10.1007/BF02392081)DOI
  4. C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 2, 507--592. doi:10.4171/JEMS/928 (https://doi.org/10.4171/JEMS/928)DOI
  5. A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116--117. doi:10.2307/1968123 (https://doi.org/10.2307/1968123)DOI
  6. N. V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Mat. Zh. 17 (1976), no. 2, 290--303 (Russian); English transl.\ Siberian Math. J. 17 (1976), no. 2, 226--236. doi:10.1007/BF00967569 (https://doi.org/10.1007/BF00967569)DOI
  7. N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161--175, 239 (Russian); English transl.\ Math. USSR-Izv. 16 (1981), no. 1, 151--164. doi:10.1070/IM1981v016n01ABEH001283 (https://doi.org/10.1070/IM1981v016n01ABEH001283)DOI
  8. E. M. Landis, Second order equations of elliptic and parabolic type , Translations of Mathematical Monographs, vol. 171, American Mathematical Society, Providence, RI, 1998.
  9. G. M. Lieberman, Second Order Parabolic Differential Equations , World Scientific, River Edge, NJ, 1996. doi:10.1142/3302 (https://doi.org/10.1142/3302)DOI
  10. A. Loher, C. Mooney and C. Mouhot, Döblin--Fourier cancellation and kinetic Aleksandrov estimates, arXiv:2610.03083 (2026).arXiv
  11. A. Loher, C. Mooney, C. Mouhot, L. Niebel, L. Silvestre and Y. Zhu, Forthcoming work.
  12. C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15--30. doi:10.1007/BF02416445 (https://doi.org/10.1007/BF02416445)DOI
  13. K. Tso, On an Aleksandrov--Bakel'man type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations 10 (1985), no. 5, 543--553. doi:10.1080/03605308508820388 (https://doi.org/10.1080/03605308508820388)DOI
  14. Y. Zhu, Stationary kinetic Krylov--Safonov estimates in space dimension one, preprint (2026), arXiv:2608.31158 (https://arxiv.org/abs/2608.31158).arXiv

Version history

  1. v1Submitted by Amélie LoherInitial depositCurrentOct 28, 2026