Barbashin-Krasovskii theorem for stochastic differential equations

Ignatyev, Oleksiy; Mandrekar, V.

doi:10.1090/S0002-9939-2010-10466-5

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.

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