= Solution
Because $f$ is a continuous probability density, there are a point $z$, a radius $r>0$, and $\epsilon>0$ such that
$$
f\geq\epsilon\quad\text{on }B(z,r).
$$
By the <recurrence of planar Brownian motion>, the smaller disc $B(z,r/2)$ is visited at arbitrarily large times. Starting anywhere in that smaller disc, Brownian continuity and compactness give a uniform probability $\delta>0$ of staying in $B(z,r)$ for a fixed time $u>0$.
Apply the <Strong Markov property> at successive visits separated by at least $u$. The conditional probability of each stay event is at least $\delta$, so the conditional <Borel-Cantelli lemma> gives infinitely many successful stays almost surely. Every success adds at least $\epsilon u$ to $A_t$. Since $A_t$ is nondecreasing,
$$
\boxed{A_t\longrightarrow\infty\quad\text{almost surely}.}
$$
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