### A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts

Bogachev V, Röckner M (2000)
Theory of Probability & Its Applications 45(3): 363-378.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Autor*in
Bogachev, V.; Röckner, MichaelUniBi
Einrichtung
Abstract / Bemerkung
Let A = (A(ij)) be a mapping with values in the space of the nonnegative symmetric operators on R-n and let B = (B-i) be a Borel vector field on R-n such that A is locally uniformly nondegenerate, A(ij) is an element of H-loc(p,1)(R-n), B-i is an element of L-loc(p)(R-n), where p > n. We show that the existence of a Lyapunov function for the operator L-A,L-B f = Sigma A(ij) partial derivative (xi) partial derivative (xj) f+Sigma B-i partial derivative (xi) f is sufficient for the existence of a probability measure mu with a strictly positive continuous density in the class H-loc(p,1)(R-n) such that mu satisfies L-A,L-B* mu = 0 in the weak sense and is an invariant measure for the diffusion with the generator L-A,L-B on domain C-o(infinity)(R-n). For arbitrary continuous nondegenerate A and locally bounded B, we prove the existence of absolutely continuous solutions. An analogous generalization of Khasminskii's theorem is obtained for manifolds.
Stichworte
invariant measure; diffusion process
Erscheinungsjahr
2000
Zeitschriftentitel
Theory of Probability & Its Applications
Band
45
Ausgabe
3
Seite(n)
363-378
ISSN
0040-585X
Page URI
https://pub.uni-bielefeld.de/record/1616433

### Zitieren

Bogachev V, Röckner M. A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts. Theory of Probability & Its Applications. 2000;45(3):363-378.
Bogachev, V., & Röckner, M. (2000). A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts. Theory of Probability & Its Applications, 45(3), 363-378. doi:10.1137/S0040585X97978348
Bogachev, V., and Röckner, M. (2000). A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts. Theory of Probability & Its Applications 45, 363-378.
Bogachev, V., & Röckner, M., 2000. A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts. Theory of Probability & Its Applications, 45(3), p 363-378.
V. Bogachev and M. Röckner, “A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts”, Theory of Probability & Its Applications, vol. 45, 2000, pp. 363-378.
Bogachev, V., Röckner, M.: A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts. Theory of Probability & Its Applications. 45, 363-378 (2000).
Bogachev, V., and Röckner, Michael. “A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts”. Theory of Probability & Its Applications 45.3 (2000): 363-378.

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