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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 $\phi$.

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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 $U$?

Yes, some books even take the definition of regular points to be that eg. see online notes "Classical potential theory and Brownian motion". In terms of conditions besides the Port-Stone book, I would look at the Garnett-Marshall book "Harmonic measure" eg. the section on the Wiener-series for regular points in terms of capacities.

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