What is known about the distribution of the hitting position of a line by a 2d brownian motion?
I've tried to make some simulations of a 2d brownian motion where every computational step has a distribution N(0,1) in every coordinate. I let it hit the line $x=1$ and I keep track of the $y$ coordinate of hitting.
Unfortunately many of the simulations requires too much steps to reach the line so I've decided to discard the simulation if the line is not reached in 30000 steps.
How much information am I losting, discarding these trajectories?
I then made 100000 simulation of trajectories hitting the line. The distribution function is similar to a gaussian, but there are here and there huge numbers too. For example, the 99,9% of the mass is in the interval [-60,60] but there are occurrences in the interval [-40000,40000].
Is likely to be a gaussian?
EDIT:
Surely is not a gaussian. This is the plot of the distribution I obtained: 
Notice the two peaks near zero. I can't figure out what they represent. Do you?
The second coordinate $Y$ of the hitting point has a symmetric Cauchy distribution. A notable feature of these distributions is that their mean do not exist.
Recall that the symmetric Cauchy distribution with parameter $a\gt0$ has density on the real line $$y\mapsto\dfrac{a}{\pi(a^2+y^2)}.$$ Indeed it is quite different from a normal density since its tail $P(|Y|\gt y)$ when $y\to+\infty$ is $\Theta\left(\frac1y\right)$.
The two peaks in the figure near $y=0$ are probably simulation artefacts, since every Cauchy distributions is unimodal.