Partial asymptotic stability in probability of stochastic differential equations

A system of stochastic differential equations dX(t)=f(t,X)dt+∑i=1kgi(t,X)dWi(t) which has a zero solution X=0X=0 is considered. It is assumed that there exists a function V(t,x)V(t,x), positive definite with respect to part of the state variables which also has the infinitesimal upper limit with respect to the part of the variables and such that the corresponding operator LVLV is nonpositive. It is proved that if the nondegeneracy condition of the matrix corresponding to the coefficients of Wiener processes holds with respect to the part of the variables with appropriate function r(x)r(x), then the zero solution is asymptotically stable in probability with respect to the part of the variables.