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

Bogachev VI, Röckner M, Stannat W (2002)
Sbornik: Mathematics 193(7-8): 945-976.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Bogachev, Vladimir I.; Röckner, MichaelUniBi; Stannat, W
Abstract / Bemerkung
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 derivative(xi)partial derivative(xj) + b(i)partial derivative(xi) in local coordinates, where a(ij) is an element of H-loc(p,1), b(i) is an element of L-loc(p) for some p > d, and the matrix (a'j) 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: integralLphif dmu = 0 for all phi is an element of C-0(infinity)(M). In addition, the uniqueness of invariant probability measures for the corresponding semigroups (T-t(mu))tgreater than or equal to0 generated by the operator L is investigated. It is proved that if a probability measure it on M satisfies the equation L*mu = 0 and (L - I) (C-0(infinity)(M)) is dense in L-1 (M,p), then it 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)(tgreater than or equal to0) generated by L has at most one invariant probability measure.
Erscheinungsjahr
2002
Zeitschriftentitel
Sbornik: Mathematics
Band
193
Ausgabe
7-8
Seite(n)
945-976
ISSN
1064-5616
Page URI
https://pub.uni-bielefeld.de/record/1613244

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Bogachev VI, Röckner M, Stannat W. Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Mathematics. 2002;193(7-8):945-976.
Bogachev, V. I., Röckner, M., & Stannat, W. (2002). Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Mathematics, 193(7-8), 945-976. https://doi.org/10.1070/SM2002v193n07ABEH000665
Bogachev, V. I., Röckner, M., and Stannat, W. (2002). Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Mathematics 193, 945-976.
Bogachev, V.I., Röckner, M., & Stannat, W., 2002. Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Mathematics, 193(7-8), p 945-976.
V.I. Bogachev, M. Röckner, and W. Stannat, “Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions”, Sbornik: Mathematics, vol. 193, 2002, pp. 945-976.
Bogachev, V.I., Röckner, M., Stannat, W.: Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Mathematics. 193, 945-976 (2002).
Bogachev, Vladimir I., Röckner, Michael, and Stannat, W. “Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions”. Sbornik: Mathematics 193.7-8 (2002): 945-976.

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