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Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 2, Pages 240–270 (Mi tvp172)  

This article is cited in 20 scientific papers (total in 20 papers)

Estimates of densities of stationary distributions and transition probabilities of diffusion processes

V. I. Bogacheva, M. Röcknerb, S. V. Shaposhnikovc

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Bielefeld University
c M. V. Lomonosov Moscow State University

Abstract: We obtain lower bounds for solutions to second order elliptic and parabolic equations on the whole space. Our method is based on the study of the dependence of a constant in Harnack's inequality on the coefficients of the equation. As an application we obtain lower bounds for densities of stationary distributions and transition probabilities of diffusion processes with unbounded drift coefficients.

Keywords: Harnack inequality, transition probabilities, stationary distribution, lower bounds for solutions to parabolic equations.


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English version:
Theory of Probability and its Applications, 2008, 52:2, 209–236

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Received: 01.12.2006

Citation: V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Estimates of densities of stationary distributions and transition probabilities of diffusion processes”, Teor. Veroyatnost. i Primenen., 52:2 (2007), 240–270; Theory Probab. Appl., 52:2 (2008), 209–236

Citation in format AMSBIB
\by V.~I.~Bogachev, M.~R\"ockner, S.~V.~Shaposhnikov
\paper Estimates of densities of stationary distributions and transition probabilities of diffusion processes
\jour Teor. Veroyatnost. i Primenen.
\yr 2007
\vol 52
\issue 2
\pages 240--270
\jour Theory Probab. Appl.
\yr 2008
\vol 52
\issue 2
\pages 209--236

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    This publication is cited in the following articles:
    1. S. V. Shaposhnikov, “On Interior Estimates of the Sobolev Norms of Solutions of Elliptic Equations”, Math. Notes, 83:2 (2008), 285–289  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Bogachev V.I., Da Prato G., Röckner M., “On parabolic equations for measures”, Comm. Partial Differential Equations, 33:3 (2008), 397–418  crossref  mathscinet  zmath  isi  elib  scopus
    3. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Positive Densities of Transition Probabilities of Diffusion Processes”, Theory Probab. Appl., 53:2 (2009), 194–215  mathnet  crossref  crossref  isi  elib
    4. V. I. Bogachev, N. V. Krylov, M. Röckner, “Elliptic and parabolic equations for measures”, Russian Math. Surveys, 64:6 (2009), 973–1078  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    5. Theory Probab. Appl., 54:1 (2010), 68–96  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. I. Noarov, “On some diffusion processes with stationary distributions”, Theory Probab. Appl., 54:3 (2010), 525–533  mathnet  crossref  crossref  mathscinet  isi
    7. Fornaro S., Fusco N., Metafune G., Pallara D., “Sharp upper bounds for the density of some invariant measures”, Proc. Roy. Soc. Edinburgh Sect. A, 139:6 (2009), 1145–1161  crossref  mathscinet  zmath  isi  elib  scopus
    8. Shaposhnikov S.V., “Lower estimates for densities of solutions to parabolic equations for measures”, Dokl. Math., 80:3 (2009), 877–881  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    9. Bogachev V.I., Röckner M., Shaposhnikov S.V., “Lower estimates of densities of solutions of elliptic equations for measures”, Dokl. Math., 79:3 (2009), 329–334  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    10. Lorenzi L., Zamboni A., “Cores for parabolic operators with unbounded coefficients”, J. Differential Equations, 246:7 (2009), 2724–2761  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. Geissert M., Lorenzi L., Schnaubelt R., “$L^p$-regularity for parabolic operators with unbounded time–dependent coefficients”, Ann. Mat. Pura Appl. (4), 189:2 (2010), 303–333  crossref  mathscinet  zmath  isi  scopus
    12. 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
    13. Aibeche A., Laidoune K., Rhandi A., “Time dependent Lyapunov functions for some Kolmogorov semigroups perturbed by unbounded potentials”, Arch. Math. (Basel), 94:6 (2010), 565–577  crossref  mathscinet  zmath  isi  scopus
    14. 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
    15. 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
    16. A. I. Noarov, “Stationary diffusion processes with discontinuous drift coefficients”, St. Petersburg Math. J., 24:5 (2013), 795–809  mathnet  crossref  mathscinet  zmath  isi  elib
    17. 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
    18. 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
    19. 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
    20. V. I. Bogachev, A. I. Kirillov, S. V. Shaposhnikov, “Distances between stationary distributions of diffusions and solvability of nonlinear Fokker–Planck–Kolmogorov equations”, Theory Probab. Appl., 62:1 (2018), 12–34  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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