Independent increments give a restart property at deterministic times. The strong Markov theorem extends it to stopping times: after any observable random time, the translated future is fresh Brownian motion and carries no information from the stopped past.
Brownian filtration
Definition. Let \[ \mathcal F_t^0=\sigma(B_s:0\le s\le t). \] The usual Brownian filtration is the completion of the right-continuous augmentation \[ \mathcal F_t=\bigcap_{u>t} (\mathcal F_u^0\vee\mathcal N), \] where \(\mathcal N\) contains the \(P\)-null events.
Theorem (strong Markov property). Let \(\tau\) be an almost surely finite stopping time. Then \[ W_t:=B_{\tau+t}-B_\tau,\qquad t\ge0, \] is Brownian motion independent of \(\mathcal F_\tau\). Equivalently, for every bounded measurable path functional \(F\) and bounded \(\mathcal F_\tau\)-measurable \(Z\), \[ \mathbb E\!\left[ ZF((B_{\tau+t}-B_\tau)_{t\ge0})\right] = \mathbb EZ\,\mathbb E F((B_t)_{t\ge0}). \]
Proof. First suppose \(\tau\) takes values in a countable deterministic set \(\{t_k\}\). For \(A\in\mathcal F_\tau\), the event \(A\cap\{\tau=t_k\}\) belongs to \(\mathcal F_{t_k}\). For any bounded continuous cylinder functional \[ F(x)=f(x(s_1),\ldots,x(s_m)), \] deterministic-time independent increments give \[ \begin{aligned} &\mathbb E[ 1_{A\cap\{\tau=t_k\}} F((B_{\tau+s}-B_\tau)_s)]\\ &\qquad= P(A\cap\{\tau=t_k\})\, \mathbb EF((B_s)_s). \end{aligned} \] Sum over \(k\). A monotone-class argument extends the identity from cylinder functionals to bounded path-measurable \(F\). For a general \(\tau\), define the dyadic approximations \[ \tau_n=2^{-n}(\lfloor2^n\tau\rfloor+1). \] Then \(\tau_n\downarrow\tau\), each \(\tau_n\) is a countably valued stopping time, and \(\mathcal F_\tau\subseteq\mathcal F_{\tau_n}\). Brownian continuity gives, for each finite set of \(s\)'s, \[ B_{\tau_n+s}-B_{\tau_n} \to B_{\tau+s}-B_\tau \quad\text{almost surely}. \] Apply dominated convergence to the countably valued identity for bounded continuous cylinder \(F\). This gives the factorization against every bounded \(Z\in\mathcal F_\tau\), hence independence and the Brownian finite-dimensional laws. Path continuity completes the identification.
Semigroup form
Corollary. For every bounded measurable \(f\), \(t\ge0\), and a.s. finite stopping time \(\tau\), \[ \mathbb E[f(B_{\tau+t})\mid\mathcal F_\tau] =P_tf(B_\tau), \] where \[ P_tf(x)=\int f(x+y)p_t(y)\,\mathrm{d} y. \]
Proof. Conditional on \(\mathcal F_\tau\), \(B_{\tau+t}=B_\tau+W_t\), where \(W_t\sim N(0,t)\) is independent of that sigma-field. Integrating over \(W_t\) gives the heat semigroup.
Restarting at a barrier
Proposition. For \(T_a=\inf\{t:B_t=a\}\), on \(\{T_a<\infty\}\), \[ (B_{T_a+t}-a)_{t\ge0} \] is standard Brownian motion independent of \(\mathcal F_{T_a}\).
Proof. Continuity gives \(B_{T_a}=a\). Apply the strong Markov theorem at \(T_a\).
Corollary (recurrence on the line). One-dimensional Brownian motion hits every fixed point almost surely and returns to every neighborhood infinitely often.
Proof. The reflection formula gives \(P(T_a<\infty)=1\). After hitting \(a\), strong Markov restarts a Brownian motion at \(a\), which hits \(a\pm\varepsilon\) almost surely. Repeating at successive exit and return times constructs infinitely many visits to the neighborhood. A countable intersection over rational centers and radii gives simultaneous recurrence.
Exit-time decomposition
Proposition. Let \(D\subset\mathbb R^d\) be open and \(\tau_D=\inf\{t:B_t\notin D\}\). On \(\{\tau_D<\infty\}\), conditional on the exit position \(B_{\tau_D}\), the future path has the law of Brownian motion started from that position and is otherwise independent of the pre-exit path.
This is the probabilistic basis of harmonic measure, domain decomposition, excursions, and regenerative arguments. It is stronger than merely conditioning on \(B_{\tau_D}\): the entire sigma-field accumulated before exit becomes irrelevant to the translated future.