New criterion of partial asymptotic stability in probability of stochastic differential equations

Autonomous system of stochastic differential equations dX(t)=f(X)dt+∑i=1kgi(X)dWi(t) which has a zero solution X=0X=0 is considered. It is assumed that there exists a positive definite with respect to part of the variables function V(X)V(X), such that corresponding operator LV   is nonpositive. It is proved that if the set M0={X:LV=0}⋂{X:V(X)>0}M0={X:LV=0}⋂{X:V(X)>0} does not include entire semitrajectories of the system (2.1) almost surely then the zero solution is asymptotically stable in probability with respect to part of the variables.