I have the following problem which I do not know how to solve - I know there is a set of results by Donsker and Varadhan, but my maths is not good enough to understand them I'm afraid. Suppose I have a Brownian motion $ W(t), 0<t<\infty$. I'd like to know if there is any formula for the almost sure behaviour of
$$ \liminf_{T\rightarrow\infty}\int_{0}^{T}\frac{1}{(1+[W(t)]^2)^{2}}dt $$
I would be really grateful for any help.