Let $\mathbb D$ be the complex unit disk. Let $B$ be a standard complex Brownian motion started at $0\in \mathbb D$. Let $\tau = \inf\{ t : B_t \in \partial\mathbb D\}$. I am trying to show that if $A\subset \mathbb D$ is a compact connected subset, then $\mathbb P(B[0,\tau] \cap A \not=\emptyset) \geq C \operatorname{diam} A$, where $\operatorname{diam}$ is the diameter, and $C$ is a constant which does not depend on $A$.
By approximating a general $A$ by the closures of open sets, we can assume that $A$ is the closure of an open set, so that $A$ contains a curve of length equal to its diameter. Hence we can instead assume $A$ is a curve if we so desire. I also tried to do this using the Beurling projection theorem (the probability that $B$ hits a set $A$ is greater than or equal to the probability that $B$ hits its radial projection, $\{|z| : z\in A\}$) but was not successful, since the radial projection can be much smaller than $A$.
Does anyone have a proof of this fact?
For whoever might stumble upon this post in the future, I've written down a detailed proof of the aforementioned result here:
https://davidpechersky.com/2020/08/02/brownian-motion-hitting-probabilities/
It is also worth mentioning that this result is an exercise from Chapter 2 of Lawler's "Conformally Invariant Processes in the Plane."