Bulletin of the
Korean Mathematical Society BKMS

We consider a sequence of stochastic differential equations which is related with a result of Khasminskii which studies the behavior of trajectory of stochastic process defined by differential equation $\frac {dx}{dt}=\epsilon F(x,t,\omega), x(0)=x_0.$ Let $Z$ be an ergodic Markov process on a separable metric space $E$, $F:R^d\times E\rightarrow R$, and $X_n, n=1,2\dots$ satisfy $dX_n(t)= nF(X_n(t),Z(n^2 t))dt.$ We show that $\{X_n\}$ is relative compact and the behavior of limit process under some conditions.

Keywords: stochastic differential equation, weak convergence

MSC numbers: primary 60H05, 60F17; secondary 60G44