Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the time for $P$ to cross a coordinate hyperplane?



Time for brownian motion to cross a coordinate plane
Similar to what @zhoraster points out in the comments, this can be boiled down to 1-dimensional Brownian motion because projecting the $n$-dimensional Brownian motion $X(t)$ onto the normal direction to the hyperplane gives a Brownian motion $B(t)$. So the question becomes about a the time it takes for $B(t)$ to reach a point $a\in \mathbb R$. The distribution is usually proved using the "reflection principle," which is explained in the following OCW lecture


[ocw.mit.edu/courses/sloan-school-of-management/15-070j-advanced-stochastic-processes-fall-2013/lecture-notes/MIT15\_070JF13\_Lec7.pdf](http://ocw.mit.edu/courses/sloan-school-of-management/15-070j-advanced-stochastic-processes-fall-2013/lecture-notes/MIT15_070JF13_Lec7.pdf)


(In particular the CDF of $T\_a = \inf\{t:B(t)=a\}$ is in the 2nd line of the proof of Proposition 3)


