Solution
= Solution
By the <Tonelli theorem> and the planar <Brownian transition density>,
$$
\mathbb E[A_t]
=\int_0^t\mathbb E[f(X_s)]\,ds.
$$
For $s\geq1$,
$$
\mathbb E[f(X_s)]
=\int_{\mathbb R^2}f(y)\frac1{2\pi s}
e^{-|y-X_0|^2/(2s)}\,dy
\leq\frac1{2\pi s},
$$
while for $0\leq s\leq1$ it is at most $\|f\|_\infty$. Hence
$$
\mathbb E[A_t]\leq\|f\|_\infty+\frac{\log t}{2\pi}
$$
for $t\geq1$, and consequently
$$
\boxed{\mathbb E[A_t/t]\longrightarrow0.}
$$