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Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 4, Pages 652–674 (Mi tvp124)  

This article is cited in 21 scientific papers (total in 21 papers)

Global regularity and estimates for solutions of parabolic equations

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: Given a second-order parabolic operator
$$ Lu(t,x):=\frac{\partial u(t,x)}{\partial t}+a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x)+b^i(t,x)\partial_{x_i}u(t,x), $$
we consider the weak parabolic equation $L^{*}\mu=0$ for Borel probability measures on $(0,1)\times\mathbf{R}^d$. The equation is understood as the equality
$$ \int_{(0,1)\timesR^d} Lu d\mu=0 $$
for all smooth functions $u$ with compact support in $(0,1)\timesR^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with $L$. We show that under broad assumptions, $\mu$ has the form $\mu=\varrho(t,x) dt dx$, where the function $x\mapsto\varrho(t,x)$ is Sobolev, $|\nabla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,\tau]\times\mathbf{R}^d$, and $\varrho\in L^p([0,\tau]\timesR^d)$ for all $p\in[1,+\infty)$ and $\tau<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,\tau]\timesR^d$ is given.

Keywords: parabolic equations for measures, transition probabilities, regularity of solutions of parabolic equations, estimates of solutions of parabolic equations.


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English version:
Theory of Probability and its Applications, 2006, 50:4, 561–581

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Citation: V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Global regularity and estimates for solutions of parabolic equations”, Teor. Veroyatnost. i Primenen., 50:4 (2005), 652–674; Theory Probab. Appl., 50:4 (2006), 561–581

Citation in format AMSBIB
\by V.~I.~Bogachev, M.~R\"ockner, S.~V.~Shaposhnikov
\paper Global regularity and estimates for solutions of parabolic equations
\jour Teor. Veroyatnost. i Primenen.
\yr 2005
\vol 50
\issue 4
\pages 652--674
\jour Theory Probab. Appl.
\yr 2006
\vol 50
\issue 4
\pages 561--581

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    This publication is cited in the following articles:
    1. V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Estimates of densities of stationary distributions and transition probabilities of diffusion processes”, Theory Probab. Appl., 52:2 (2008), 209–236  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Bogachev V.I., Da Prato G., Röckner M., Stannat W., “Uniqueness of solutions to weak parabolic equations for measures”, Bull. Lond. Math. Soc., 39:4 (2007), 631–640  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. Spina Ch., “Kernel estimates for a class of Kolmogorov semigroups”, Arch. Math. (Basel), 91:3 (2008), 265–279  crossref  mathscinet  zmath  isi  scopus
    5. 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
    6. 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
    7. Theory Probab. Appl., 54:1 (2010), 68–96  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. 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
    9. 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
    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. 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
    17. Angiuli L., Lorenzi L., “On the Dirichlet and Neumann Evolution Operators in”, Potential Anal., 41:4 (2014), 1079–1110  crossref  mathscinet  zmath  isi  scopus
    18. Kusuoka S., “Holder Continuity and Bounds For Fundamental Solutions To Nondivergence Form Parabolic Equations”, Anal. PDE, 8:1 (2015), 1–32  crossref  mathscinet  zmath  isi  scopus
    19. 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
    20. 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
    21. Kunze M., Lorenzi L., Rhandi A., “Kernel estimates for nonautonomous Kolmogorov equations”, Adv. Math., 287 (2016), 600–639  crossref  mathscinet  zmath  isi  elib  scopus
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