I want to compute the hitting probability of a bounded plane by a Brownian motion starting at the origin. In other words, given the coordinates of a quadrilateral A , can we compute $P(T_{A}<\infty)$? How can I go about it?

First I try the specific case where A is centered at the x-axis and is parallel to the zx plane. Also, let A be a rectangle.

Then, $\{T_{A} <\infty\}=\{B_{1}(t)=a, |B_{2}(t)|\leq b,|B_{3}(t)|\leq c$ for some $t>0\}$. Here by a,b,c I mean the distance from origin, length and width of the rectangle $A$.

So I have to compute: $P_{0}\{(B_{1}(t)=a)\cap (|B_{2}(t)|\leq b)\cap (|B_{3}(t)|\leq c)$ for some $t>0\}$

Can I use the independence of the coordinates of a Brownian motion?

![Bounded Plane][1] [1]: https://i.sstatic.net/2bqqg.png