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? I claim no, because the above events require a common t.

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

Update

Can I someone solve for the above A. My friend advised me to use Fourier transforms.

$$\nabla^2\Phi(\vec{r})=-\delta(\vec{r}),\;\;\Phi(\vec{r})=0\;\;\text{for}\;\;\vec{r}\in A,$$

$$P_A=\int_A \frac{\partial\Phi}{\partial\vec{r}}\cdot\hat{n}\;dS,$$

with $\hat{n}$ a unit vector normal to $A$ and pointing outward.