|B| -distribution (Brownian Motion)

[Bridgeport]

Let $B(t)=(B_1(t),...,B_n(t))$ be $\mathbb R^n $valued Brownian motion. I need to find distribution of |B(t)| \, and \, |B(t)|^2

I can use brute force i.e.$P(|B(t)|^2\leq \alpha)= \int_{\mathbb R^n}I_{(<x,x>\leq \alpha)}fdx$, where $f$is Gaussian density, but I wonder if there is more elegant way?

Ogromnoe spasibo!

[Хорхе]

Try to Google for Bessel process.

[Henrylee]

If $B_i(t)$ are independent processes look at $\chi$ and $\chi^2$ distributions.