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 $\phi$.



Brownian motion hitting open set starting from its boundary

> 
> 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"](https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s6_v3_article-09.pdf). In terms of conditions besides the [Port-Stone book](https://www.sciencedirect.com/book/9780125618502/brownian-motion-and-classical-potential-theory), I would look at the Garnett-Marshall book "Harmonic measure" eg. the section on the Wiener-series for regular points in terms of capacities.


