Proof of positive probability of exit of Brownian Motion through an open subset of the boundary of a compact connected $\mathbb{R}^2$ set.

Question

Suppose that $D$ is a subset of $\mathbb{R}^2$ with a smooth connected boundary $\partial D$. Let $U$ be an open subset of $\partial D$ and $x$ be a point in the interior of $D$. How do we know that the probability of exit of Brownian Motion, starting at $x$, through $U$ (first hitting time) is positive?

Edit: Can we use the fundamental solution to Laplace's equation for the disk and transform the problem (via a conformal mapping) to the region? Seems to work if the region has no holes.