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Mat. Sb., 2002, Volume 193, Number 7, Pages 3–36 (Mi msb665)  

This article is cited in 49 scientific papers (total in 49 papers)

Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions

V. I. Bogacheva, M. Röcknerb, W. Stannatb

a M. V. Lomonosov Moscow State University
b Bielefeld University

Abstract: Let $M$ be a complete connected Riemannian manifold of dimension $d$ and let $L$ be a second order elliptic operator on $M$ that has a representation $L=a^{ij}\partial_{x_i}\partial_{x_j}+b^i\partial_{x_i}$ in local coordinates, where $a^{ij}\in H^{p,1}_{\mathrm{loc}}$, $b^i\in L^p_{loc}$ for some $p>d$, and the matrix $(a^{ij})$ is non-singular. The aim of the paper is the study of the uniqueness of a solution of the elliptic equation $L^*\mu=0$ for probability measures $\mu$, which is understood in the weak sense: $\displaystyle\int L\varphi f d\mu=0$ for all $\varphi\in C_0^\infty(M)$. In addition, the uniqueness of invariant probability measures for the corresponding semigroups $(T_t^\mu)_{t\geqslant 0}$ generated by the operator $L$ is investigated. It is proved that if a probability measure $\mu$ on $M$ satisfies the equation $L^*\mu=0$ and $(L-I)(C^\infty_0(M))$ is dense in $L^1(M,\mu)$, then $\mu$ is a unique solution of this equation in the class of probability measures. Examples are presented (even with $a^{ij}=\delta^{ij}$ and smooth $b^i$) in which the equation $L^*\mu=0$ has more than one solution in the class of probability measures. Finally, it is shown that if $p>d+2$, then the semigroup $(T_t)_{t\geqslant 0}$ generated by $L$ has at most one invariant probability measure.


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English version:
Sbornik: Mathematics, 2002, 193:7, 945–976

Bibliographic databases:

UDC: 517.956+517.98+519.2
MSC: 58J05, 47F05
Received: 08.01.2002

Citation: V. I. Bogachev, M. Röckner, W. Stannat, “Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions”, Mat. Sb., 193:7 (2002), 3–36; Sb. Math., 193:7 (2002), 945–976

Citation in format AMSBIB
\by V.~I.~Bogachev, M.~R\"ockner, W.~Stannat
\paper Uniqueness of solutions of elliptic equations and
uniqueness of invariant measures of diffusions
\jour Mat. Sb.
\yr 2002
\vol 193
\issue 7
\pages 3--36
\jour Sb. Math.
\yr 2002
\vol 193
\issue 7
\pages 945--976

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    This publication is cited in the following articles:
    1. V. I. Bogachev, M. Röckner, “On $L^p$-uniqueness of symmetric diffusion operators on Riemannian manifolds”, Sb. Math., 194:7 (2003), 969–978  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Bogachev V.I., Da Prato G., Röckner M., Sobol Z., “Global gradient bounds for dissipative diffusion operators”, C. R. Math. Acad. Sci. Paris, 339:4 (2004), 277–282  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. Bogachev V.I., Röckner M., Wang Feng-Yu, “Invariance implies Gibbsian: Some new results”, Comm. Math. Phys., 248:2 (2004), 335–355  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. Bogachev V.I., Da Prato G., Röckner M., “Existence of solutions to weak parabolic equations for measures”, Proc. London Math. Soc. (3), 88:3 (2004), 753–774  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    5. V. I. Bogachev, N. V. Krylov, M. Röckner, “Regularity and global bounds of densities of invariant measures of diffusion processes”, Dokl. Math., 72:3 (2005), 934–938  mathnet  mathscinet  zmath  isi  elib  elib
    6. Bogachev V.I., Krylov N.V., Röckner M., “Elliptic equations for measures: Regularity and global bounds of densities”, J. Math. Pures Appl. (9), 85:6 (2006), 743–757  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Bogachev V.I., Da Prato G., Röckner M., Stannat W., “Uniqueness of solutions to weak parabolic equations for measures”, Bull. Lond. Math. Soc., 39 (2007), 631–640  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    8. 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
    9. S. V. Shaposhnikov, “The nonuniqueness of solutions to elliptic equations for probability measures”, Dokl. Math., 77:3 (2008), 401–403  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    10. Shaposhnikov S.V., “On nonuniqueness of solutions to elliptic equations for probability measures”, J. Funct. Anal., 254:10 (2008), 2690–2705  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
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