Uspekhi Matematicheskikh Nauk
General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 2009, Volume 64, Issue 6(390), Pages 5–116 (Mi umn9326)  

This article is cited in 71 scientific papers (total in 71 papers)

Elliptic and parabolic equations for measures

V. I. Bogacheva, N. V. Krylovb, M. Röcknerc

a M. V. Lomonosov Moscow State University
b University of Minnesota, Minneapolis, USA
c Bielefeld University, Germany

Abstract: This article gives a detailed account of recent investigations of weak elliptic and parabolic equations for measures with unbounded and possibly singular coefficients. The existence and differentiability of densities are studied, and lower and upper bounds for them are discussed. Semigroups associated with second-order elliptic operators acting in $L^p$-spaces with respect to infinitesimally invariant measures are investigated.
Bibliography: 181 titles.

Keywords: elliptic equation, parabolic equation, stationary distribution of a diffusion process, transition probability.


Full text: PDF file (1460 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2009, 64:6, 973–1078

Bibliographic databases:

UDC: 517.987+517.972
MSC: 35R05, 35R15, 35J15, 35K10, 58J65, 60J60
Received: 05.10.2009

Citation: V. I. Bogachev, N. V. Krylov, M. Röckner, “Elliptic and parabolic equations for measures”, Uspekhi Mat. Nauk, 64:6(390) (2009), 5–116; Russian Math. Surveys, 64:6 (2009), 973–1078

Citation in format AMSBIB
\by V.~I.~Bogachev, N.~V.~Krylov, M.~R\"ockner
\paper Elliptic and parabolic equations for measures
\jour Uspekhi Mat. Nauk
\yr 2009
\vol 64
\issue 6(390)
\pages 5--116
\jour Russian Math. Surveys
\yr 2009
\vol 64
\issue 6
\pages 973--1078

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Tian R., Wei J., Tang Ya., “the Dirichlet Problem For Nonlocal Elliptic Equations”, Appl. Anal.  crossref  isi
    2. Bogachev V.I., Roeckner M., Shaposhnikov S.V., “on the Ambrosio-Figalli-Trevisan Superposition Principle For Probability Solutions to Fokker-Planck-Kolmogorov Equations”, J. Dyn. Differ. Equ.  crossref  isi
    3. Manita O.A., Shaposhnikov S.V., “Nonlinear parabolic equations for measures”, Dokl. Math., 86:3 (2012), 857–860  crossref  mathscinet  zmath  isi  elib  scopus
    4. Bogachev V.I., Korolev A.V., Pilipenko A.Yu., “Non uniform averagings in the ergodic theorem for stochastic flows”, Dokl. Math., 81:3 (2010), 422–425  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    5. Shaposhnikov S.V., “Estimates of Solutions of Parabolic Equations for Measures”, Dokl. Math., 82:2 (2010), 769–772  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. Bogachev V.I., Kirillov A.I., Shaposhnikov S.V., “Invariant Measures of Diffusions with Gradient Drifts”, Dokl. Math., 82:2 (2010), 790–793  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. Bogachev V.I., Röckner M., Shaposhnikov S.V., “On uniqueness problems related to elliptic equations for measures”, J. Math. Sci., 176:6 (2011), 759–773  crossref  mathscinet  zmath  elib  scopus
    8. Bogachev V.I., Kirillov A.I., Shaposhnikov S.V., “On probability and integrable solutions to the stationary Kolmogorov equation”, Dokl. Math., 83:3 (2011), 309–313  crossref  mathscinet  zmath  isi  elib  elib  scopus
    9. Agafontsev B.V., Bogachev V.I., Shaposhnikov S.V., “A condition for the positivity of the density of an invariant measure”, Dokl. Math., 83:3 (2011), 332–336  crossref  mathscinet  zmath  isi  elib  elib  scopus
    10. Tonoyan L.G., “Nonlinear elliptic equations for measures”, Dokl. Math., 84:1 (2011), 558–561  crossref  mathscinet  zmath  isi  elib  elib  scopus
    11. Shaposhnikov S.V., “On the uniqueness of integrable and probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equations”, Dokl. Math., 84:1 (2011), 565–570  crossref  mathscinet  zmath  isi  elib  elib  scopus
    12. S. V. Shaposhnikov, “On the uniqueness of a probabilistic solution of the Cauchy problem for the Fokker–Planck–Kolmogorov equation”, Theory Probab. Appl., 56:1 (2012), 96–115  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. S. V. Shaposhnikov, “Regular and qualitative properties of solutions for parabolic equations for measures”, Theory Probab. Appl., 56:2 (2011), 252–279  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. Li Y., Dwivedi G., Huang W., Kemp M.L., Yi Y., “Quantification of degeneracy in biological systems for characterization of functional interactions between modules”, J. Theoret. Biol., 302 (2012), 29–38  crossref  mathscinet  zmath  isi  elib  scopus
    15. Bogachev V.I., Kirillov A.I., Shaposhnikov S.V., “Integrable solutions of the stationary Kolmogorov equation”, Dokl. Math., 85:3 (2012), 309–314  crossref  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    16. Bogachev V.I., Röckner M., Shaposhnikov S.V., “On positive and probability solutions to the stationary Fokker-Planck-Kolmogorov equation”, Dokl. Math., 85:3 (2012), 350–354  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    17. V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Journal of Mathematical Sciences, 213:3 (2016), 287–459  mathnet  crossref  mathscinet
    18. Dzh. O. Ogun, Yu. N. Orlov, V. Zh. Sakbaev, “O preobrazovanii prostranstva nachalnykh dannykh dlya zadachi Koshi s osobennostyami resheniya tipa vzryva”, Preprinty IPM im. M. V. Keldysha, 2012, 087, 31 pp.  mathnet
    19. Ren J., Wu J., “On regularity of invariant measures of multivalued stochastic differential equations”, Stochastic Process. Appl., 122:1 (2012), 93–105  crossref  mathscinet  zmath  isi  elib  scopus
    20. V. Zh. Sakbaev, “On the properties of ambiguity and irreversibility of dynamical maps of the initial data space of Cauchy problem”, P-Adic Num Ultrametr Anal Appl, 4:4 (2012), 306  crossref  mathscinet  zmath  scopus
    21. Evans S.N., Ralph P.L., Schreiber S.J., Sen A., “Stochastic population growth in spatially heterogeneous environments”, J. Math. Biol., 66:3 (2013), 423–476  crossref  mathscinet  zmath  isi  elib  scopus
    22. Bogachev V.I., Röckner M., Shaposhnikov S.V., “On parabolic inequalities for generators of diffusions with jumps”, Probab. Theory Relat. Fields, 2013  crossref  mathscinet  isi  elib  scopus
    23. O. A. Manita, S. V. Shaposhnikov, “Nonlinear parabolic equations for measures”, St. Petersburg Math. J., 25:1 (2014), 43–62  mathnet  crossref  mathscinet  zmath  isi  elib
    24. V. N. Denisov, “Stabilization conditions for the solution of the Cauchy problem for a parabolic equation with growing lower order coefficients”, Dokl. Math., 87:3 (2013), 348  crossref  mathscinet  zmath  isi  elib  scopus
    25. V. Zh. Sakbaev, “Gradient blow-up of solutions to the Cauchy problem for the Schrödinger equation”, Proc. Steklov Inst. Math., 283 (2013), 165–180  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    26. M. N. Lukintsova, “The radon transform in an infinite-dimensional space with Gaussian measure”, Dokl. Math., 88:3 (2013), 674  crossref  mathscinet  zmath  isi  scopus
    27. S. V. Shaposhnikov, “The Fokker–Planck–Kolmogorov equations with a potential and a non-uniformly elliptic diffusion matrix”, Trans. Moscow Math. Soc., 74 (2013), 15–29  mathnet  crossref  mathscinet  zmath  elib
    28. Bogachev V.I., Röckner M., Shaposhnikov S.V., “On uniqueness of solutions to the Cauchy problem for degenerate Fokker-Planck-Kolmogorov equations”, J. Evol. Equ., 13:3 (2013), 577–593  crossref  mathscinet  zmath  isi  scopus
    29. V. I. Bogachev, A. I. Kirillov, S. V. Shaposhnikov, “A stationary Fokker-Planck-Kolmogorov equation with a potential”, Dokl. Math., 89:1 (2014), 24–29  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    30. M. Kh. Numan Elsheikh, D. O. Ogun, Yu. N. Orlov, R. V. Pleshakov, V. Zh. Sakbaev, “Usrednenie sluchainykh polugrupp i neodnoznachnost kvantovaniya gamiltonovykh sistem”, Preprinty IPM im. M. V. Keldysha, 2014, 019, 28 pp.  mathnet
    31. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “On existence of Lyapunov functions for a stationary Kolmogorov equation with a probability solution”, Dokl. Math., 90:1 (2014), 424  crossref  crossref  mathscinet  mathscinet  zmath  isi  elib  scopus
    32. V. I. Bogachev, A. Yu. Veretennikov, S. V. Shaposhnikov, “Differentiability of invariant measures of diffusions with respect to a parameter”, Dokl. Math, 91:1 (2015), 76  crossref  mathscinet  zmath  isi  scopus
    33. O. A. Manita, S. V. Shaposhnikov, “Fokker-Planck-Kolmogorov equations with a potential term on a domain”, Dokl. Math, 91:1 (2015), 26  crossref  mathscinet  zmath  isi  scopus
    34. O.A.. Manita, S.V.. Shaposhnikov, “On the Cauchy Problem for Fokker–Planck–Kolmogorov Equations with Potential Terms on Arbitrary Domains”, J Dyn Diff Equat, 2015  crossref  mathscinet  scopus
    35. O. A. Manita, M. S. Romanov, S. V. Shaposhnikov, “Uniqueness of probability solutions to nonlinear Fokker-Planck-Kolmogorov equation”, Dokl. Math, 91:2 (2015), 142  crossref  mathscinet  zmath  isi  scopus
    36. Wen Huang, Min Ji, Zhenxin Liu, Yingfei Yi, “Steady States of Fokker–Planck Equations: I. Existence”, J Dyn Diff Equat, 2015  crossref  mathscinet  isi  scopus
    37. Wen Huang, Min Ji, Zhenxin Liu, Yingfei Yi, “Steady States of Fokker–Planck Equations: II. Non-existence”, J Dyn Diff Equat, 2015  crossref  mathscinet  isi  scopus
    38. L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, Theoret. and Math. Phys., 185:2 (2015), 1582–1598  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    39. Manita O.A. Romanov M.S. Shaposhnikov S.V., “on Uniqueness of Solutions To Nonlinear Fokker-Planek-Kolmogorov Equations”, 128, 2015, 199–226  crossref  mathscinet  zmath  isi  scopus
    40. Feehan P.M.N., Pop C.A., “on the Martingale Problem For Degenerate-Parabolic Partial Differential Operators With Unbounded Coefficients and a Mimicking Theorem For It(O)Over-Cap Processes”, 367, no. 11, 2015, 7565–7593  crossref  mathscinet  zmath  isi  scopus
    41. Huang W., Ji M., Liu Zh., Yi Y., “Integral Identity and Measure Estimates For Stationary Fokker-Planck Equations”, 43, no. 4, 2015, 1712–1730  crossref  mathscinet  zmath  isi  scopus
    42. Bogachev V.I. Da Prato G. Roeckner M. Shaposhnikov S.V., “An Analytic Approach To Infinite-Dimensional Continuity and Fokker-Planck-Kolmogorov Equations”, 14, no. 3, 2015, 983–1023  mathscinet  zmath  isi
    43. Kusuoka S., “Holder Continuity and Bounds For Fundamental Solutions To Nondivergence Form Parabolic Equations”, 8, no. 1, 2015, 1–32  crossref  mathscinet  zmath  isi  scopus
    44. Huang W., Ji M., Liu Zh., Yi Y., “Steady States of Fokker-Planck Equations: III. Degenerate Diffusion”, 28, no. 1, 2016, 127–141  crossref  mathscinet  zmath  isi  scopus
    45. Huang W., Ji M., Liu Zh., Yi Y., “Stochastic Stability of Measures in Gradient Systems”, 314, 2016, 9–17  crossref  mathscinet  zmath  isi  scopus
    46. Bogachev V.I. Roeckner M. Shaposhnikov S.V., “Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations”, J. Funct. Anal., 271:5 (2016), 1262–1300  crossref  mathscinet  zmath  isi  scopus
    47. Li Ya., Yi Y., “Systematic Measures of Biological Networks I: Invariant Measures and Entropy”, Commun. Pure Appl. Math., 69:9 (2016), 1777–1811  crossref  mathscinet  zmath  isi  elib  scopus
    48. Puhalskii A.A., “On large deviations of coupled diffusions with time scale separation”, Ann. Probab., 44:4 (2016), 3111–3186  crossref  mathscinet  zmath  isi  elib  scopus
    49. Bogachev V.I., Shaposhnikov S.V., Veretennikov A.Yu., “Differentiability of solutions of stationary Fokker–Planck–Kolmogorov equations with respect to a parameter”, Discret. Contin. Dyn. Syst., 36:7 (2016), 3519–3543  crossref  mathscinet  zmath  isi  elib  scopus
    50. Bogachev V.I. Roeckner M. Shaposhnikov S.V., “Estimates of distances between transition probabilities of diffusions”, Dokl. Math., 93:2 (2016), 135–139  crossref  mathscinet  zmath  isi  elib  scopus
    51. Kunze M., Lorenzi L., Rhandi A., “Kernel estimates for nonautonomous Kolmogorov equations”, Adv. Math., 287 (2016), 600–639  crossref  mathscinet  zmath  isi  elib  scopus
    52. V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, Theoret. and Math. Phys., 191:3 (2017), 886–909  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    53. Bogachev V.I., Shaposhnikov S.V., “Integrability and Continuity of Solutions to Fokker-Planck-Kolmogorov Equations”, Dokl. Math., 96:3 (2017), 583–586  crossref  mathscinet  zmath  isi  scopus
    54. Wang F.-Yu., “Integrability Conditions For SDEs and Semilinear SPDEs”, Ann. Probab., 45:5 (2017), 3223–3265  crossref  mathscinet  zmath  isi  scopus
    55. Huang W., Ji M., Liu Zh., Yi Y., “Concentration and Limit Behaviors of Stationary Measures”, Physica D, 369 (2018), 1–17  crossref  mathscinet  isi  scopus
    56. Zhang X., “Stochastic Hamiltonian Flows With Singular Coefficients”, Sci. China-Math., 61:8 (2018), 1353–1384  crossref  mathscinet  isi  scopus
    57. V. I. Bogachev, “Ornstein–Uhlenbeck operators and semigroups”, Russian Math. Surveys, 73:2 (2018), 191–260  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    58. Wang F.-Yu., “Estimates For Invariant Probability Measures of Degenerate Spdes With Singular and Path-Dependent Drifts”, Probab. Theory Relat. Field, 172:3-4 (2018), 1181–1214  crossref  mathscinet  zmath  isi  scopus
    59. Bogachev V.I., Popova S.N., Shaposhnikov S.V., “On Sobolev Classes Containing Solutions to Fokker-Planck-Kolmogorov Equations”, Dokl. Math., 98:2 (2018), 498–501  crossref  zmath  isi  scopus
    60. Masoero M., “On the Long Time Convergence of Potential Mfg”, NoDea-Nonlinear Differ. Equ. Appl., 26:2 (2019), 15  crossref  mathscinet  isi  scopus
    61. Bogachev I V., Popova S.N., Shaposhnikov V S., “On Sobolev Regularity of Solutions to Fokker-Planck-Kolmogorov Equations With Drifts in l-1”, Rend. Lincei-Mat. Appl., 30:1 (2019), 205–221  crossref  mathscinet  zmath  isi  scopus
    62. Chen L., Dong Zh., Jiang J., Niu L., Zhai J., “Decomposition Formula and Stationary Measures For Stochastic Lotka-Volterra System With Applications to Turbulent Convection”, J. Math. Pures Appl., 125 (2019), 43–93  crossref  mathscinet  zmath  isi  scopus
    63. D. V. Zavadsky, V. Zh. Sakbaev, “Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure”, Proc. Steklov Inst. Math., 306 (2019), 102–119  mathnet  crossref  crossref  mathscinet  isi  elib
    64. Ji M., Shen Zh., Yi Y., “Quantitative Concentration of Stationary Measures”, Physica D, 399 (2019), 73–85  crossref  isi
    65. Ji M., Shen Zh., Yi Y., “Convergence to Equilibrium in Fokker-Planck Equations”, J. Dyn. Differ. Equ., 31:3, SI (2019), 1591–1615  crossref  isi
    66. Ji M., Qi W., Shen Zh., Yi Y., “Existence of Periodic Probability Solutions to Fokker-Planck Equations With Applications”, J. Funct. Anal., 277:11 (2019), UNSP 108281  crossref  isi
    67. Li Ya., “a Data-Driven Method For the Steady State of Randomly Perturbed Dynamics”, Commun. Math. Sci., 17:4 (2019), 1045–1059  isi
    68. Xie L., Zhang X., “Ergodicity of Stochastic Differential Equations With Jumps and Singular Coefficients”, Ann. Inst. Henri Poincare-Probab. Stat., 56:1 (2020), 175–229  crossref  isi
    69. V. I. Bogachev, “Non-uniform Kozlov–Treschev averagings in the ergodic theorem”, Russian Math. Surveys, 75:3 (2020), 393–425  mathnet  crossref  crossref  mathscinet  isi  elib
    70. Bogachev I V., Shaposhnikov V S., “Representations of Solutions to Fokker-Planck-Kolmogorov Equations With Coefficients of Low Regularity”, J. Evol. Equ., 20:2 (2020), 355–374  crossref  mathscinet  isi
    71. V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, “On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation”, Sb. Math., 212:6 (2021), 745–781  mathnet  crossref  crossref  isi  elib
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:1757
    Full text:538
    First page:68

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021