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Teoriya Veroyatnostei i ee Primeneniya, 2017, Volume 62, Issue 1, Pages 16–43
DOI: https://doi.org/10.4213/tvp5094 (Mi tvp5094) 		 
This article is cited in 8 scientific papers (total in 8 papers)

Distances between stationary distributions of diffusions and solvability of nonlinear Fokker–Planck–Kolmogorov equations

V. I. Bogachevabc, A. I. Kirillovd, S. V. Shaposhnikovabc

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b St. Tikhon's Orthodox University, Moscow
c National Research University "Higher School of Economics" (HSE), Moscow
d Russian Foundation for Basic Research, Moscow
Full-text PDF (513 kB) Citations (8)
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DOI: https://doi.org/10.4213/tvp5094
Abstract: This paper is concerned with investigation of stationary distributions of diffusion processes. We obtain estimates for the Kantorovich, Prohorov, and total variation distances between stationary distributions of diffusions with different diffusion matrices and different drift coefficients. Applications are given to nonlinear stationary Fokker–Planck–Kolmogorov equations, for which new conditions for the existence and uniqueness of probability solutions are found; moreover, these conditions are optimal in a sense.
Keywords: stationary Fokker–Planck–Kolmogorov equation, total variation distance, Kantorovich metric, Prohorov metric, nonlinear Fokker–Planck–Kolmogorov equation.
Funding agency Grant number
Russian Science Foundation 14-11-00196
This work was supported by the Russian Science Foundation at Moscow State University (grant 14-11-00196).
Received: 28.06.2016
Revised: 28.07.2016
Accepted: 10.10.2016
English version:
Theory of Probability and its Applications, 2018, Volume 62, Issue 1, Pages 12–34
DOI: https://doi.org/10.1137/S0040585X97T988460
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: V. I. Bogachev, A. I. Kirillov, S. V. Shaposhnikov, “Distances between stationary distributions of diffusions and solvability of nonlinear Fokker–Planck–Kolmogorov equations”, Teor. Veroyatnost. i Primenen., 62:1 (2017), 16–43; Theory Probab. Appl., 62:1 (2018), 12–34
Citation in format AMSBIB
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\paper Distances between stationary distributions of diffusions and solvability of nonlinear Fokker--Planck--Kolmogorov equations
\jour Teor. Veroyatnost. i Primenen.
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\pages 16--43
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  • https://www.mathnet.ru/eng/tvp/v62/i1/p16
    • This publication is cited in the following 8 articles:
    1. Alexander Y. Mitrophanov, “The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective”, Mathematics, 12:11 (2024), 1608  crossref
    2. V. I. Bogachev, S. V. Shaposhnikov, “Nonlinear Fokker–Planck–Kolmogorov equations”, Russian Math. Surveys, 79:5 (2024), 751–805  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    3. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Kolmogorov problems on equations for stationary and transition probabilities of diffusion processes”, Theory Probab. Appl., 68:3 (2023), 342–369  mathnet  crossref  crossref
    4. W. Tang, Y. P. Zhang, X. Y. Zhou, “Exploratory HJB equations and their convergence”, SIAM J. Control Optim., 60:6 (2022), 3191  crossref  mathscinet
    5. V. I. Bogachev, M. Roeckner, S. V. Shaposhnikov, “Convergence in variation of solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures”, J. Funct. Anal., 276:12 (2019), 3681–3713  crossref  mathscinet  zmath  isi  scopus
    6. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “On Convergence to Stationary Distributions for Solutions of Nonlinear Fokker–Planck–Kolmogorov Equations”, J Math Sci, 242:1 (2019), 69  crossref  mathscinet
    7. V. I. Bogachev, A. F. Miftakhov, S. V. Shaposhnikov, “Differential properties of semigroups and estimates of distances between stationary distributions of diffusions”, Dokl. Math., 99:2 (2019), 175–180  crossref  mathscinet  isi
    8. V. I. Bogachev, M. Roeckner, S. V. Shaposhnikov, “Convergence to stationary measures in nonlinear Fokker–Planck–Kolmogorov equations”, Dokl. Math., 98:2 (2018), 452–457  mathnet  crossref  crossref  zmath  isi  elib  scopus
    Citing articles in Google Scholar: Russian citations, English citations
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