Independent Brownian arms at a deterministic time
= Independent Brownian arms at a deterministic time
For a deterministic $h>0$, the processes $B_{h-t}-B_h$, $0\leq t\leq h$, and $B_{h+t}-B_h$, $t\geq0$, are independent standard <Brownian motions>. The first uses reversed, sign-changed past increments; the second uses future increments independent of the entire past. The first process is not generally independent of $B_h$, but translating a range by $B_h$ preserves its <Lebesgue measure> pathwise.