Consider a Brownian particle $B(t)$ in $\mathbb{R}^2$ starting at the origin, and let $U$ be an open set in $\mathbb{R}^2$. For a positive real number $\tau$, define $$ P_1(\tau) = \text{ probability that } B(\tau) \in U, $$ and $$ P_2(\tau) = \text{ probability that } B(t) \in U \text{ for some } t \leq \tau. $$ Clearly, one has that $P_1(\tau) \leq P_2(\tau)$ by simple inclusion.
My question is, if in addition one assumes that the motion is recurrent (as on the plane), do we also have some comparability relation like $$ P_2(\tau) \leq c P_1(\tau), $$ where $c$ is a constant independent of $\tau$? Any hints will be highly appreciated.
Additional remark: one can easily get comparability upto constants dependent on $\tau$, but this does not require recurrence. For the question above, it seems not so difficult to (naively!) believe that a comparability may hold if $\tau \geq \text{dist}(0, U)^2$.