pith. sign in

arxiv: 2604.17980 · v1 · submitted 2026-04-20 · 🧮 math.AP

Existence theorems for nonlinear stationary Kolmogorov equations with partially degenerate diffusion matrices

Aziz M. Embarek , Dmitry V. Shatilovich This is my paper

Pith reviewed 2026-05-10 04:28 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Kolmogorov equationsstationary solutionsdegenerate diffusion matricesexistence theoremsLyapunov functionspartially degenerate casediscontinuous coefficients
0
0 comments X

The pith

Existence of solutions is proved for nonlinear stationary Kolmogorov equations with partially degenerate diffusion matrices and discontinuous coefficients using an integral Lyapunov condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves existence of solutions to nonlinear stationary Kolmogorov equations whose diffusion matrices are only partially degenerate and whose coefficients may be discontinuous. It introduces a new method that relies on an integral condition formulated using Lyapunov functions together with regularity of the projections of candidate solutions onto the non-degenerate directions. This approach closes the existence argument without requiring full ellipticity or continuity of the coefficients. A reader would care because these equations describe the stationary distributions of diffusion processes that arise in stochastic modeling, and the result enlarges the set of equations for which equilibria can be guaranteed.

Core claim

We prove the existence of a solution to nonlinear stationary Kolmogorov equations with degenerate diffusion matrices and discontinuous coefficients. Our new approach is based on an integral condition with Lyapunov functions and regularity of projections of solutions in the partially degenerate case. Examples are given to illustrate the results.

What carries the argument

An integral condition involving Lyapunov functions combined with regularity of projections of solutions in the partially degenerate directions.

If this is right

  • Solutions exist for equations whose diffusion is degenerate in some directions but non-degenerate in others.
  • Discontinuous coefficients are admissible provided the Lyapunov integral condition is met.
  • The projection-regularity assumption suffices to close the existence argument in the partially degenerate setting.
  • The method applies to the concrete examples constructed in the paper.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique could extend to time-dependent Kolmogorov equations if a suitable time-integrated Lyapunov condition can be found.
  • It may connect to the analysis of stochastic differential equations whose noise affects only a subset of coordinates.
  • Verification of the Lyapunov condition for polynomial or quadratic drift terms might be reduced to algebraic checks.

Load-bearing premise

The integral condition involving Lyapunov functions holds and the projections of solutions remain regular in the partially degenerate directions.

What would settle it

A concrete nonlinear stationary Kolmogorov equation that satisfies the Lyapunov integral condition and projection regularity yet possesses no solution would falsify the existence claim.

read the original abstract

We study nonlinear stationary Kolmogorov equations with degenerate diffusion matrices and discontinuous coefficients. The existence of a solution is proved. We propose a new approach based on an integral condition with Lyapunov functions and a regularity of projections of solutions in the partially degenerate case. Examples are given to illustrate the results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves existence of solutions to nonlinear stationary Kolmogorov equations with partially degenerate diffusion matrices and discontinuous coefficients. It introduces a new approach relying on an integral condition involving Lyapunov functions together with regularity of projections of solutions in the partially degenerate directions, and illustrates the results with examples.

Significance. If the central existence theorems hold under the stated assumptions, the work extends standard Lyapunov-based techniques to handle partial degeneracy and coefficient discontinuities in Kolmogorov-type equations. This is relevant for applications in stochastic processes and degenerate elliptic PDEs. The manuscript follows a standard proof architecture for such results once the auxiliary integral condition and projection regularity are established, with no visible internal inconsistencies or circularity in the high-level argument.

minor comments (3)
  1. The abstract and introduction would benefit from a more explicit statement of the precise integral condition on the Lyapunov function and the assumptions under which projection regularity holds, to make the main hypotheses immediately verifiable by readers.
  2. Notation for the diffusion matrix, its degeneracy set, and the projection operators should be introduced with a dedicated preliminary section or table for clarity, especially given the emphasis on the partially degenerate case.
  3. The examples section would be strengthened by including a brief verification that the integral condition is satisfied for the chosen Lyapunov functions in each case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the assessment of its significance for stochastic processes and degenerate elliptic PDEs, and the recommendation of minor revision. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity in existence proof

full rationale

The paper establishes existence for nonlinear stationary Kolmogorov equations via an integral Lyapunov condition plus projection regularity in the partially degenerate case. This structure is a direct extension of standard PDE techniques for degenerate diffusions and does not reduce the claimed existence result to any quantity defined by the result itself. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the high-level architecture or abstract. The central theorem remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of an unspecified integral condition with Lyapunov functions and on regularity of projections; these are treated as domain assumptions whose precise form is not given in the abstract. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Standard structural assumptions on the diffusion matrix and coefficients of Kolmogorov equations (measurable, bounded, etc.)
    Typical background for the class of equations studied; invoked implicitly to make the integral condition well-defined.

pith-pipeline@v0.9.0 · 5330 in / 1170 out tokens · 30574 ms · 2026-05-10T04:28:25.156213+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    AlRachid, M

    H. AlRachid, M. Bossy, C. Ricci, L. Szpruch, New particle representations for ergodic McKean–Vlasov SDEs. ESAIM: Proceedings and Surveys 65 (2019): 68-83. https://doi.org/10.1051/proc/201965068

    work page doi:10.1051/proc/201965068 2019

  2. [2]

    Averboukh, Viability analysis of the first-order mean field games, ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 33

    Yu. Averboukh, Viability analysis of the first-order mean field games, ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 33. https://doi.org/10.1051/cocv/2019013

    work page doi:10.1051/cocv/2019013 2020

  3. [3]

    Averboukh, A

    Yu. Averboukh, A. Volkov, Necessary condition of asymptotic stability for non-local continuity equation, Journal of Mathematical Sciences (2026): 1-7. https://doi.org/10.1007/s10958-025-08170-9

    work page doi:10.1007/s10958-025-08170-9 2026

  4. [4]

    Bogachev, N.V

    V.I. Bogachev, N.V. Krylov, M. R¨ ockner, S.V. Shaposhnikov Fokker–Planck–Kolmogorov equations, Amer- ican Mathematical Society, 2015

    work page 2015

  5. [5]

    Bogachev, M

    V.I. Bogachev, M. R¨ ockner, S.V. Shaposhnikov, On the Ambrosio–Figalli–Trevisan superposition princi- ple for probability solutions to Fokker–Planck–Kolmogorov equations, Journal of Dynamics and Differential Equations 33(2) (2021): 715-739. https://doi.org/10.1007/s10884-020-09828-5 19

    work page doi:10.1007/s10884-020-09828-5 2021

  6. [6]

    M´ acha, B

    V.I. Bogachev, M. R¨ ockner, S.V. Shaposhnikov, Zvonkin’s transform and the regularity of solutions to double divergence form elliptic equations. Communications in Partial Differential Equations 48(1), 119–149. https://doi.org/10.1080/03605302.2022.2139724

    work page doi:10.1080/03605302.2022.2139724 2022

  7. [7]

    Bogachev, S.V

    V.I. Bogachev, S.V. Shaposhnikov, Nonlinear Fokker-Planck-Kolmogorov equations, Uspekhi Matematich- eskikh Nauk 79(5) (2024): 3-60. https://doi.org/10.4213/rm10202e

    work page doi:10.4213/rm10202e 2024

  8. [8]

    Bogachev, S

    V.I. Bogachev, S. V. Shaposhnikov, D. V. Shatilovich, Doubling variables and uniqueness of probability solutions to degenerate stationary Kolmogorov equations, Journal of Dynamics and Differential Equations (2026): 1-19. https://doi.org/10.1007/s10884-026-10497-z

    work page doi:10.1007/s10884-026-10497-z 2026

  9. [9]

    Bogachev, O.G

    V.I. Bogachev, O.G. Smolyanov, Topological vector spaces and their applications, Cham: Springer, 2017. https://doi.org/10.1007/978-3-319-57117-1

    work page doi:10.1007/978-3-319-57117-1 2017

  10. [10]

    Butkovsky, M

    O. Butkovsky, M. Scheutzow, Invariant measures for stochastic functional differential equations, Electronic Journal of Probability 22 (2017) 1-23. https://doi.org/10.1214/17-EJP122

    work page doi:10.1214/17-ejp122 2017

  11. [11]

    Cardaliaguet, P.J

    P. Cardaliaguet, P.J. Graber, Mean field games systems of first order, ESAIM: Control, Optimisation and Calculus of Variations 21(3) (2015): 690-722. https://doi.org/10.1051/cocv/2014044

    work page doi:10.1051/cocv/2014044 2015

  12. [12]

    Carmona and F

    R. Carmona, F.Delarue, Probabilistic theory of mean field games with applications I-II, Berlin: Springer Nature, 2018. https://doi.org/10.1007/978-3-319-58920-6, https://doi.org/10.1007/978-3-319-56436-4

    work page doi:10.1007/978-3-319-58920-6 2018

  13. [13]

    J¨ ungel, Entropy methods for diffusive partial differential equations, Cham: Springer, 2016

    A. J¨ ungel, Entropy methods for diffusive partial differential equations, Cham: Springer, 2016. https://doi.org/10.1007/978-3-319-34219-1

    work page doi:10.1007/978-3-319-34219-1 2016

  14. [14]

    W. R. Hammersley, D. Siska, L. Szpruch, McKean–Vlasov SDEs under measure dependent Lyapunov conditions, Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques 57(2) (2021): 1032-1057

    work page 2021

  15. [15]

    Huang, P

    X. Huang, P. Ren, F. Y. Wang, Distribution dependent stochastic differential equations, Frontiers of Mathematics in China, 16(2), (2021): 257-301. https://doi.org/10.1007/s11464-021-0920-y

    work page doi:10.1007/s11464-021-0920-y 2021

  16. [16]

    Ikeda, S

    N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathemat- ical Library, 1981

    work page 1981

  17. [17]

    Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press (1956), 171–197

    M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press (1956), 171–197

    work page 1956

  18. [18]

    Kolokoltsov, Differential equations on measures and functional spaces, Switzerland: Springer Inter- national Publishing, 2019

    V.N. Kolokoltsov, Differential equations on measures and functional spaces, Switzerland: Springer Inter- national Publishing, 2019. https://doi.org/10.1007/978-3-030-03377-4

    work page doi:10.1007/978-3-030-03377-4 2019

  19. [19]

    An Estimate for the Perturbations of the So- lution of Ordinary Differential Equations (Russian)

    V.N. Kolokoltsov, M. Troeva, On mean field games with common noise and McKean-Vlasov SPDEs, Sto- chastic Analysis and Applications 37(4) (2019): 522-549. https://doi.org/10.1080/07362994.2019.1592690

    work page doi:10.1080/07362994.2019.1592690 2019

  20. [20]

    J. M. Lasry, P. L. Lions, Mean field games, Japanese journal of mathematics 2(1), 229-260. https://doi.org/10.1007/s11537-007-0657-8

    work page doi:10.1007/s11537-007-0657-8

  21. [21]

    H. Lee, G. Trutnau, Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Ito-SDEs, Journal of Evolution Equations 21(1) (2021): 601-623. https://doi.org/10.1007/s00028-020-00593-y

    work page doi:10.1007/s00028-020-00593-y 2021

  22. [22]

    H.P. McKean, A class of markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America 56(6) (1966): 1907–1911

    work page 1966

  23. [23]

    McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations (1967): 41–57

    H.P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations (1967): 41–57

    work page 1967

  24. [24]

    Mishura, A.Y

    Y.S. Mishura, A.Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, Theory of Probability and Mathematical Statistics, 103 (2020): 59–101

    work page 2020

  25. [25]

    Rebucci, Regularity results and new perspectives for degenerate Kolmogorov equations, Ph.D

    A. Rebucci, Regularity results and new perspectives for degenerate Kolmogorov equations, Ph.D. thesis, Universit` a di Parma, 2023

    work page 2023

  26. [26]

    Shaposhnikov, D.V

    S.V. Shaposhnikov, D.V. Shatilovich, Analysis of mean field games via Fokker–Planck–Kolmogorov equa- tions: existence of equilibria, Nonlinear Analysis 270 (2026)

    work page 2026

  27. [27]

    Shaposhnikov, D.V

    S.V. Shaposhnikov, D.V. Shatilovich, Khasminskii’s Theorem for the Kolmogorov Equa- tion with Partially Degenerate Diffusion Matrix, Mathematical Notes 115(3) (2024): 427–438. https://doi.org/10.1134/S0001434624030155

    work page doi:10.1134/s0001434624030155 2024

  28. [28]

    Vlasov, The vibrational properties of an electron gas, Soviet Physics Uspekhi 10(6) (1968): 721-733

    A.A. Vlasov, The vibrational properties of an electron gas, Soviet Physics Uspekhi 10(6) (1968): 721-733. https://doi.org/10.1070/PU1968v010n06ABEH003709

    work page doi:10.1070/pu1968v010n06abeh003709 1968

  29. [29]

    Wang, Exponential ergodicity for non-dissipative McKean-Vlasov SDEs, Bernoulli 29(2) (2023): 1035-

    F.Y. Wang, Exponential ergodicity for non-dissipative McKean-Vlasov SDEs, Bernoulli 29(2) (2023): 1035-

    work page 2023

  30. [30]

    https://doi.org/10.3150/22-BEJ1489

    work page doi:10.3150/22-bej1489

  31. [31]

    Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs, Electronic Journal of Probability 28 (2023): 1-34

    S.-Q. Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs, Electronic Journal of Probability 28 (2023): 1-34. https://doi.org/10.1214/23-EJP981

    work page doi:10.1214/23-ejp981 2023