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For a stopping time $T$ of a Brownian filtration, define the reflected stochastic processThe Brownian reflection at a stopping time form of the Brownian reflection principle states that $\hat{B}$ is again standard Brownian motion. If $T=\infty$, leave the path unchanged; the second branch is used only when $T<t$.

For an almost surely finite $T$, the Strong Markov property says that $(B_{T+s}-B_T{)}_{s\ge 0}$ is a fresh Brownian motion independent of ${\mathcal{F}}_T$. Its negative has the same probability distribution, so reflecting the future preserves the full path probability distribution. Allowing $T=\infty$ follows by applying this argument to $T\wedge n$ and restricting to each fixed finite time interval.