Limit distributions of a solution of a stochastic diffusion equation

The process $\xi(t)$ being a solution of the stochastic diffusion equation (1), $0<t\le1$, the limit distribution of the process $T^{-1/2}\mathrm g(\xi(tT))$, where
$$ \mathrm g(x)=\int_0^x\exp\Bigl\{-2\int_0^u\frac{a(v)}{\sigma^2(v)}\,dv\Bigr\}\,du, $$
as $T\to\infty$ is considered.