I am trying to find the distribution of first hitting times of a 2D biased random walk to reach a set of points in $\mathbb{R}^2$. I found the answer for the 1D case of brownian motion here https://en.wikipedia.org/wiki/First-hitting-time_model but there are no references and no work to get to that result. I would like suggestions about how to do it for my case or references on this matter.
The formula I would like to find should be similar to the one for the 1D case
$f(t)\equiv\frac{|x_c-x_0|}{\sqrt{4\pi Dt^3}} \exp\left(- \frac{(x_c-x_0)^2}{4Dt}\right)$
Let $B_t = (B_t^x, B_t^y)$ be a 2-dimensional Brownian motion. In particular $B_t^x$ and $B_t^y$ are independent $1$-dimensional Brownian motions. Define the first hitting time of the line $L_{x_0} = \{(x,y) \in \mathbb{R}^2 \, : \,x=x_0 \}$ by $$\tau = \inf\{ t>0 \, : \, B_t \in L_c\} = \inf\{ t>0 \, : \, B_t^x = x_0\}$$ From the one-dimensional result it follows that $$f_\tau(t)\equiv\frac{|x_c-x_0|}{\sqrt{4\pi Dt^3}} \exp\left(- \frac{(x_c-x_0)^2}{4Dt}\right)$$