Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$.
Which conditions on $U$ are necessary and sufficient for $ \mathbb{P}\{\exists t\in[0,1]:W(t) \in U\}=1 $?
Most likely, there are no simple conditions on $U$, but maybe it can be somehow reformulated in terms of potential theory?
For example, is it true that $\mathbb{P}\{\exists t\in[0,1]:W(t) \in U\}=1$ if and only if $0$ is a regular point for $U$? The definition of a regular point is as in https://encyclopediaofmath.org/wiki/Potential_theory : we can assume for simplicity that $U$ is bounded, then regularity of $0$ means for each continuous function $\phi$ on $\partial U$ $\lim_{x \to 0} H_{\phi}(x)=\phi(0),\,x\in U$, where $H_{\phi}(x)$ is the generalized solution for the Dirichlet problem on $U$ with boundary values $\phi$.
