the strong Markov property of Brownian motion

Brownian motion B_t is memoryless at every FIXED time: given B_t, the future increments B_(t+s) - B_t are a fresh Brownian motion independent of the past. The deep question is whether this restart still works when the time is chosen by the process itself — a random stopping time tau, such as the first time B hits a level a. The strong Markov property answers yes, and it is the single engine behind almost every fine result about Brownian paths: the reflection principle, first-passage laws, excursion theory, and the potential theory all amount to restarting at a cleverly chosen tau.

Precisely, let tau be a stopping time for the right-continuous, completed Brownian filtration and let F_tau be the pre-tau sigma-algebra. On {tau < infinity} define the post-tau process W_s = B_(tau+s) - B_tau. The strong Markov property says W is a standard Brownian motion, independent of F_tau. The proof is the discretisation argument: approximate tau from above by stopping times tau_n taking finitely or countably many values, where the ordinary (fixed-time) Markov property applies directly; then let tau_n decrease to tau and use the almost-sure continuity of the paths to pass to the limit. Continuity of sample paths is exactly what makes the limit clean — this is why Brownian motion, a Feller process with continuous paths, is automatically strong Markov.

Why it matters: the fixed-time Markov property is genuinely weaker, and there exist Markov processes that are NOT strong Markov, so this is a theorem to be earned, not assumed. For Brownian motion the payoff is enormous. Reflecting the post-tau path (which is legal because it is a fresh symmetric Brownian motion) gives P(max_(u<=t) B_u >= a) = 2 P(B_t >= a). Restarting after a hit underlies the recursive structure of excursions away from zero and the renewal of the process at each level crossing. Without the strong form, none of these random-time arguments would be valid.