Recall that a domain $D \subseteq \mathbb C$ is called *regular* if for each point $x \in \partial D$, we have $\mathbf P\_x\lbrack \tau\_D = 0\rbrack = 1$, where $\tau\_D = \inf\{t > 0 : B\_t \notin D\}$, and $(B\_t)\_{t \ge 0}$ is a Brownian motion.


1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
2. Is every simply connected domain regular?



Is every simply connected domain regular?
Yes to both. Both were already answered here <https://math.stackexchange.com/questions/4389689/is-every-simply-connected-set-in-the-plane-regular-for-brownian-motion>


So for Q1 we can use the simple-arc criterion.



> 
> The regularity of every boundary point of an open set in $\mathbb R^2$ is in fact strongly related to connectedness. However, that's more a property of the complement of the set: Problem 4.2.16 in [1](https://www.sciencedirect.com/science/article/pii/S0022247X18303433) which is:
> 
> 
> 



> 
> *Let $D\subset\mathbb R^2$ be open, and suppose that $a\in\partial D$ has the property that there exists a point $b\not=a$ in $\mathbb R^2\setminus D\,,$ and a simple arc in $\mathbb R^2\setminus D$ connecting $a$ to $b\,.$ Show that $a$ is regular.*
> 
> 
> 



> 
> The solution is provided in [1](https://www.sciencedirect.com/science/article/pii/S0022247X18303433) section 4.5. The unit disc minus the line segment $[0,1)\times\{0\}$ clearly satisfies the properties of $D$ in that problem.
> 
> 
> 



> 
> [1](https://www.sciencedirect.com/science/article/pii/S0022247X18303433) I. Karatzas, S. Shreve, *Brownian Motion and Stochastic Calculus.*
> 
> 
> 


More generally for Q2, see Bass "probabilistic techniques in analysis" Prop. II.1.14
(or the article mentioned in the comments
["A remark on the probabilistic solution of the Dirichlet problem
for simply connected domains in the plane"](https://www.sciencedirect.com/science/article/pii/S0022247X18303433))


where they show that: *The Dirichlet problem is solvable for any simply connected domain in $\mathbb{C}$.*


