Limit behavior of functionals of solutions of diffusion type equations

The asymptotic behavior as $T \to \infty$ of the functionals $I (tT)$ with an appropriate normalizing factor is studied, where $I (t) = F (\xi (t)) + \int _ {0} ^ {t} g (\xi (s)) dW (s)$, $t \ge 0$, $F$ is a continuous function, $g$ is a locally square integrable function, $\xi$ is an unstable solution of the Itô stochastic differential equation $d \xi (t) = a (\xi (t)) dt + dW (t)$, and $a$ is a measurable and bounded function. We find the normalizing factor for the weak convergence of stochastic processes $I(tT)$, $t\ge 0$, for certain classes of these equations. The explicit form of the limit processes is established.