Regularity of solutions to Fokker–Planck–Kolmogorov equations

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Abstract

A survey of the regularity properties of solutions to Fokker-Planck-Kolmogorov equations of elliptic and parabolic type is presented. Conditions for the existence of the densities of solutions and some local properties of such densities, such as boundedness, continuity, Hölder continuity, and Sobolev continuity are discussed, as well as some global properties such as estimates on the whole space, high integrability, and membership in Sobolev classes in the whole space. New results on properties of solutions in the case of low regularity of coefficients are also presented.

About the authors

Vladimir Igorevich Bogachev

Lomonosov Moscow State University; National Research University Higher School of Economics

Email: vibogach@mail.ru
ORCID iD: 0000-0001-5249-2965
Scopus Author ID: 7005751293
ResearcherId: P-6316-2016
Doctor of physico-mathematical sciences, Professor

Stanislav Valer'evich Shaposhnikov

Lomonosov Moscow State University; National Research University Higher School of Economics

Email: starticle@mail.ru
ORCID iD: 0000-0002-3281-7061

References

  1. F. Asadian, “Regularity of measures induced by solutions of infinite dimensional stochastic differential equations”, Rocky Mount. J. Math., 27:2 (1997), 387–423
  2. P. Bauman, “Equivalence of the Green's functions for diffusion operators in ${mathbf R}^n$: a counterexample”, Proc. Amer. Math. Soc., 91:1 (1984), 64–68
  3. V. I. Bogachev, G. Da Prato, M. Röckner, “Regularity of invariant measures for a class of perturbed Ornstein–Uhlenbeck operators”, NoDEA Nonlinear Differential Equations Appl., 3:2 (1996), 261–268
  4. V. I. Bogachev, N. Krylov, M. Röckner, “Regularity of invariant measures: the case of non-constant diffusion part”, J. Funct. Anal., 138:1 (1996), 223–242
  5. V. I. Bogachev, N. V. Krylov, M. Röckner, “Elliptic regularity and essential self-adjointness of Dirichlet operators on $mathbb R^n$”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:3 (1997), 451–461
  6. V. I. Bogachev, N. V. Krylov, M. Röckner, “On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions”, Comm. Partial Differential Equations, 26:11-12 (2001), 2037–2080
  7. V. I. Bogachev, N. V. Krylov, M. Röckner, “Elliptic equations for measures: regularity and global bounds of densities”, J. Math. Pures Appl. (9), 85:6 (2006), 743–757
  8. V. I. Bogachev, I. I. Malofeev, S. V. Shaposhnikov, “On dependence of solutions to Fokker–Planck–Kolmogorov equations on their coefficients and initial data”, Math. Notes, 116:3 (2024), 421–431
  9. V. I. Bogachev, S. N. Popova, S. V. Shaposhnikov, “On Sobolev regularity of solutions to Fokker–Planck–Kolmogorov equations with drifts in $L^1$”, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30:1 (2019), 205–221
  10. V. I. Bogachev, M. Röckner, “Regularity of invariant measures on finite and infinite dimensional spaces and applications”, J. Funct. Anal., 133:1 (1995), 168–223
  11. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “The Poisson equation and estimates for distances between stationary distributions of diffusions”, J. Math. Sci. (N. Y.), 232:3 (2018), 254–282
  12. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Zvonkin's transform and the regularity of solutions to double divergence form elliptic equations”, Comm. Partial Differential Equations, 48:1 (2023), 119–149
  13. V. I. Bogachev, M. Röckner, Feng-Yu Wang, “Elliptic equations for invariant measures on finite and infinite dimensional manifolds”, J. Math. Pures Appl. (9), 80:2 (2001), 177–221
  14. V. I. Bogachev, A. V. Shaposhnikov, S. V. Shaposhnikov, “Log-Sobolev-type inequalities for solutions to stationary Fokker–Planck–Kolmogorov equations”, Calc. Var. Partial Differential Equations, 58:5 (2019), 176, 16 pp.
  15. V. I. Bogachev, S. V. Shaposhnikov, “Integrability and continuity of solutions to double divergence form equations”, Ann. Mat. Pura Appl. (4), 196:5 (2017), 1609–1635
  16. V. I. Bogachev, S. V. Shaposhnikov, “Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity”, J. Evol. Equ., 20:2 (2020), 355–374
  17. V. I. Bogachev, S. V. Shaposhnikov, “On reconstruction of coefficients of Fokker–Planck–Kolmogorov equations”, J. Evol. Equ., 25:1 (2025), 20, 18 pp.
  18. V. I. Bogachev, S. V. Shaposhnikov, A. Yu. Veretennikov, “Differentiability of solutions of stationary Fokker–Planck–Kolmogorov equations with respect to a parameter”, Discrete Contin. Dyn. Syst., 36:7 (2016), 3519–3543
  19. S. Bonaccorsi, M. Fuhrman, “Regularity results for infinite dimensional diffusions. A Malliavin calculus approach”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 10:1 (1999), 35–45
  20. S. Bonaccorsi, M. Fuhrman, “Integration by parts and smoothness of the law for a class of stochastic evolution equations”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7:1 (2004), 89–129
  21. F. Chiarenza, M. Frasca, P. Longo, “$W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients”, Trans. Amer. Math. Soc., 336:2 (1993), 841–853
  22. Jongkeun Choi, Hongjie Dong, Dong-ha Kim, Seick Kim, Regularity of elliptic equations in double divergence form and applications to Green's function estimates, 2025 (v1 – 2024), 31 pp.
  23. G. Da Prato, A. Debussche, “Absolute continuity of the invariant measures for some stochastic PDEs”, J. Stat. Phys., 115:1-2 (2004), 451–468
  24. G. Da Prato, A. Debussche, B. Goldys, “Some properties of invariant measures of non symmetric dissipative stochastic systems”, Probab. Theory Related Fields, 123:3 (2002), 355–380
  25. G. Da Prato, H. Frankowska, “Existence, uniqueness, and regularity of the invariant measure for a class of elliptic degenerate operators”, Differential Integral Equations, 17:7-8 (2004), 737–750
  26. G. Da Prato, M. Röckner, Feng-Yu Wang, “Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups”, J. Funct. Anal., 257:4 (2009), 992–1017
  27. G. Da Prato, J. Zabczyk, “Regular densities of invariant measures in Hilbert spaces”, J. Funct. Anal., 130:2 (1995), 427–449
  28. G. Da Prato, J. Zabczyk, Ergodicity for infinite dimensional systems, London Math. Soc. Lecture Note Ser., 229, Cambridge Univ. Press, Cambridge, 1996, xii+339 pp.
  29. E. Di Nezza, G. Palatucci, E. Valdinoci, “Hitchhiker's guide to the fractional Sobolev spaces”, Bull. Sci. Math., 136:5 (2012), 521–573
  30. Hongjie Dong, L. Escauriaza, Seick Kim, “On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: part II”, Math. Ann., 370:1-2 (2018), 447–489
  31. Hongjie Dong, L. Escauriaza, Seick Kim, “On $C^{1/2,1}$, $C^{1,2}$, and $C^{0,0}$ estimates for linear parabolic operators”, J. Evol. Equ., 21:4 (2021), 4641–4702
  32. Hongjie Dong, Dong-ha Kim, Seick Kim, “The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients”, Math. Ann., 392:1 (2025), 573–618
  33. Hongjie Dong, Dong-ha Kim, Seick Kim, The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: the two-dimensional case, 2025, 23 pp.
  34. Hongjie Dong, Dong-ha Kim, Seick Kim, Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form, 2025, 27 pp.
  35. Hongjie Dong, Seick Kim, “On $C^1$, $C^2$, and weak type-$(1, 1)$ estimates for linear elliptic operators”, Comm. Partial Differential Equations, 42:3 (2017), 417–435
  36. Hongjie Dong, Seick Kim, Sungjin Lee, “Note on Green's functions of non-divergence elliptic operators with continuous coefficients”, Proc. Amer. Math. Soc., 151:5 (2023), 2045–2055
  37. Hongjie Dong, Seick Kim, Sungjin Lee, “Estimates for fundamental solutions of parabolic equations in non-divergence form”, J. Differential Equations, 340 (2022), 557–591
  38. Hongjie Dong, Seick Kim, B. Sirakov, Hopf–Oleinik lemma for elliptic equations in double divergence form, 2025, 29 pp.
  39. B. Dyda, R. L. Frank, “Fractional Hardy–Sobolev–Maz'ya inequality for domains”, Studia Math., 208:2 (2012), 151–166
  40. A. Es-Sarhir, “Sobolev regularity of invariant measures for generalized Ornstein–Uhlenbeck operators”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9:4 (2006), 595–606
  41. X. Fernique, “Comparaison de mesures gaussiennes et de mesures produit”, Ann. Inst. H. Poincare Probab. Statist., 20:2 (1984), 165–175
  42. M. Fuhrman, “Regularity properties of transition probabilities in infinite dimensions”, Stochastics Stochastics Rep., 69:1-2 (2000), 31–65
  43. M. Fuhrman, “Logarithmic derivatives of invariant measure for stochastic differential equations in Hilbert spaces”, Stochastics Stochastics Rep., 71:3-4 (2001), 269–290
  44. B. Gaveau, J.-M. Moulinier, “Regularite des mesures et perturbations stochastiques de champs de vecteurs sur des espaces de dimension infinie”, Publ. Res. Inst. Math. Sci., 21:3 (1985), 593–616
  45. B. Goldys, B. Maslowski, “Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's”, Ann. Probab., 34:4 (2006), 1451–1496
  46. F. Gozzi, “Smoothing properties of nonlinear transition semigroups: case of Lipschitz nonlinearities”, J. Evol. Equ., 6:4 (2006), 711–743
  47. I. Gyöngy, Seick Kim, “Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients”, J. Differential Equations, 412 (2024), 857–880
  48. Sukjung Hwang, Seik Kim, “Green's function for second order elliptic equations in non-divergence form”, Potential Anal., 52:1 (2020), 27–39
  49. Seick Kim, Recent progress on second-order elliptic and parabolic equations in double divergence form, 2025, 20 pp.
  50. Seick Kim, Longjuan Xu, “Green's function for second order parabolic equations with singular lower order coefficients”, Commun. Pure Appl. Anal., 21:1 (2022), 1–21
  51. N. V. Krylov, “Parabolic and elliptic equations with VMO coefficients”, Comm. Partial Differential Equations, 32:1-3 (2007), 453–475
  52. N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Grad. Stud. Math., 96, Amer. Math. Soc., Providence, RI, 2008, xviii+357 pp.
  53. N. V. Krylov, “Elliptic equations with VMO $a,bin L_d$, and $cin L_{d/2}$”, Trans. Amer. Math. Soc., 374:4 (2021), 2805–2822
  54. N. V. Krylov, “Elliptic equations in Sobolev spaces with Morrey drift and the zeroth-order coefficients”, Trans. Amer. Math. Soc., 376:10 (2023), 7329–7351
  55. N. V. Krylov, “On parabolic equations in Morrey spaces with VMO $a$ and Morrey $b$, $c$”, NoDEA Nonlinear Differential Equations Appl., 32:1 (2025), 9, 20 pp.
  56. M. Kunze, L. Lorenzi, A. Rhandi, “Kernel estimates for nonautonomous Kolmogorov equations”, Adv. Math., 287 (2016), 600–639
  57. M. Kunze, M. Porfido, A. Rhandi, “Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms”, Discrete Contin. Dyn. Syst. Ser. S, 17:5-6 (2024), 2141–2172
  58. K. Laidoune, G. Metafune, D. Pallara, A. Rhandi, “Global properties of transition kernels associated with second-order elliptic operators”, Parabolic problems, Progr. Nonlinear Differential Equations Appl., 80, Birkäuser/Springer Basel AG, Basel, 2011, 415–432
  59. Haesung Lee, “Strong uniqueness of finite-dimensional Dirichlet operators with singular drifts”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 26:3 (2023), 2350009, 17 pp.
  60. Haesung Lee, “Local elliptic regularity for solutions to stationary Fokker–Planck equations via Dirichlet forms and resolvents”, Bound. Value Probl., 2025 (2025), 68, 30 pp.
  61. Haesung Lee, “Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms”, J. Math. Anal. Appl., 548:2 (2025), 129425, 30 pp.
  62. Haesung Lee, G. Trutnau, “Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Itô-SDEs”, J. Evol. Equ., 21:1 (2021), 601–623
  63. Haesung Lee, G. Trutnau, “Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space”, J. Math. Anal. Appl., 507:1 (2022), 125778, 31 pp.
  64. R. Leitão, E. A. Pimentel, M. S. Santos, “Geometric regularity for elliptic equations in double-divergence form”, Anal. PDE, 13:4 (2020), 1129–1144
  65. L. Lorenzi, A. Rhandi, Semigroups of bounded operators and second-order elliptic and parabolic partial differential equations, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2021, xxii+480 pp.
  66. P. Malliavin, Stochastic analysis, Grundlehren Math. Wiss., 313, Springer-Verlag, Berlin, 1997, xii+343 pp.
  67. O. Manita, “Positivity of transition probabilities of infinite-dimensional diffusion processes on ellipsoids”, Theory Stoch. Process., 20:2 (2015), 85–96
  68. S. Menozzi, A. Pesce, X. Zhang, “Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift”, J. Differential Equations, 272 (2021), 330–369
  69. G. Metafune, D. Pallara, A. Rhandi, “Global properties of invariant measures”, J. Funct. Anal., 223:2 (2005), 396–424
  70. G. Metafune, D. Pallara, A. Rhandi, “Global properties of transition probabilities of singular diffusions”, Теория вероятн. и ее примен., 54:1 (2009), 116–148
  71. J.-M. Moulinier, “Absolue continuite de probabilites de transition par rapport à une mesure gaussienne dans un espace de Hilbert”, J. Funct. Anal., 64:2 (1985), 275–295
  72. D. Ornstein, “A non-inequality for differential operators in the $L_1$ norm”, Arch. Ration. Mech. Anal., 11:1 (1962), 40–49
  73. S. Pagliarani, G. Lucertini, A. Pascucci, “Optimal regularity for degenerate Kolmogorov equations in non-divergence form with rough-in-time coefficients”, J. Evol. Equ., 23:4 (2023), 69, 37 pp.
  74. S. Peszat, J. Zabczyk, “Strong Feller property and irreducibility for diffusions on Hilbert spaces”, Ann. Probab., 23:1 (1995), 157–172
  75. E. A. Pimentel, Elliptic regularity theory by approximation methods, London Math. Soc. Lecture Note Ser., 477, Cambridge Univ. Press, Cambridge, 2022, xi+190 pp.
  76. I. Shigekawa, “Existence of invariant measures of diffusions on an abstract Wiener space”, Osaka J. Math., 24:1 (1987), 37–59
  77. P. Sjögren, “On the adjoint of an elliptic linear differential operator and its potential theory”, Ark. Mat., 11:1-2 (1973), 153–165
  78. P. Sjögren, “Harmonic spaces associated with adjoints of linear elliptic operators”, Ann. Inst. Fourier (Grenoble), 25:3-4 (1975), 509–518
  79. K. Suzuki, “Regularity and stability of invariant measures for diffusion processes under synthetic lower Ricci curvature bounds”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 23:2 (2022), 745–808

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