## Barbashin-Krasovskii theorem for stochastic differential equations

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- by Oleksiy Ignatyev and V. Mandrekar PDF
- Proc. Amer. Math. Soc.
**138**(2010), 4123-4128 Request permission

## Abstract:

A system of stochastic differential equations $dX(t)=f(X)dt+ \sum _{i=1}^{k}g_i(X)dW_i(t)$ which has a zero solution $X=0$ is considered. It is assumed that there exists a positive definite function $V(x)$ such that the corresponding operator $LV$ is nonpositive. It is proved that if the set $\{M:~ LV=0\}$ does not include entire semitrajectories of the system almost surely, then the zero solution is asymptotically stable in probability.## References

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## Additional Information

**Oleksiy Ignatyev**- Affiliation: Department of Statistics and Probability, Michigan State University, A408 Wells Hall, East Lansing, Michigan 48824-1027
- Email: ignatyev@stt.msu.edu
**V. Mandrekar**- Affiliation: Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824-1027
- Email: mandrekar@stt.msu.edu
- Received by editor(s): August 7, 2009
- Received by editor(s) in revised form: February 27, 2010
- Published electronically: July 7, 2010
- Communicated by: Richard C. Bradley
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**138**(2010), 4123-4128 - MSC (2010): Primary 60H10, 93E15
- DOI: https://doi.org/10.1090/S0002-9939-2010-10466-5
- MathSciNet review: 2679634