Brownian time reversal on a finite interval ($R_s=B_T-B_{T-s}$)

For standard Brownian motion and deterministic $T>0$, the process $R_s=B_T-B_{T-s}$, $0\le s\le T$, is standard Brownian motion on that interval in its own natural filtration. Disjoint reversed time intervals give independent centered normal increments with the correct variances. This is different from an arbitrary random-time shift, which can depend on future data. It is also different from the time inversion of Brownian motion transformation $t{B}_{1/t}$.

For a deterministic $h>0$, the processes $B_{h-t}-B_h$, $0\le t\le h$, and $B_{h+t}-B_h$, $t\ge 0$, 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.