Divergence of a positive planar Brownian occupation integral
= Divergence of a positive planar Brownian occupation integral
If $f$ is a continuous probability density on $\mathbb R^2$ and $B$ is planar <Brownian motion>, then
$$
\int_0^tf(B_s)\,ds\longrightarrow\infty
$$
almost surely. The density is bounded below on some disc, recurrence supplies infinitely many visits to a smaller concentric disc, and the <Strong Markov property> gives infinitely many visits of a fixed positive duration.