I need some help with the geometrical aspect of a Brownian motion and his hitting time.
If $W$ a Brownian motion and $\tau=inf { t\ge 0 \text{ st } W\_t > a } $ with $a \ge 0 $.
Can someone please draw the process $W\_{\tau}$. Is this a stopped process meanings that after $a$ the process is stopped for ever or the process is stopped (ie: $W\_s=a$ for $s<t$ such that $W\_s< a$) but then if $W\_t > a $ the process is "back to normal".

Can we say that $W\_t=W\_{min(t,\tau)}+W\_{min(t,\sigma)}$ with $\sigma= inf { t\ge 0, W\_t \ge a}$ ???

I don't know if it's clear.

Thanks