the strong Markov property in continuous time
The ordinary Markov property says: given the present, the future is independent of the past. For a continuous-time process X_t it reads E[f(X_(t+s)) given F_t] = (P_s f)(X_t), where F_t is the history up to the deterministic time t. The strong Markov property asks whether the same memorylessness survives if we replace the fixed time t by a random time tau that the process itself chooses — a stopping time, such as the first time X hits a level. This is not automatic, and it is exactly what makes the deep theory of Brownian motion and diffusions work.
Precisely: let tau be a stopping time with respect to the (right-continuous, completed) filtration, and let F_tau be the pre-tau sigma-algebra. The process is strong Markov if, on the event {tau < infinity}, the post-tau process s -> X_(tau+s) is again Markov with the SAME transition function, started afresh from X_tau and conditionally independent of F_tau. In semigroup notation, E[f(X_(tau+s)) given F_tau] = (P_s f)(X_tau) almost surely on {tau < infinity}. The fixed-time identity does not by itself force the random-time identity; one proves it by approximating tau from above by discrete-valued stopping times tau_n (which inherit the property from the ordinary Markov property) and passing to the limit using right-continuity of paths and (typically) the Feller continuity of P_s.
Why it matters: the reflection principle for Brownian motion, the distribution of first-passage times, excursion theory, Dynkin's formula, and the optional-stopping use of the martingale problem all rely on restarting the process at a random time. A Markov process need not be strong Markov — pathological examples exist — but Feller processes (right-continuous paths plus the Feller property of the semigroup) always are. The usual conditions on the filtration (right-continuity and completeness) are the standard hypothesis under which stopping-time arguments are clean.
Let B_t be Brownian motion and tau = inf{ t : B_t = a } the first hitting time of level a. The strong Markov property says B_(tau+s) - a is a fresh Brownian motion independent of F_tau. Combining this with the symmetry of that fresh motion gives the reflection principle and hence P( max_(u<=t) B_u >= a ) = 2 P( B_t >= a ).
Restarting at the random hitting time tau is exactly the strong Markov property; ordinary Markovianity at fixed times is not enough.
Strong Markov is strictly stronger than Markov: there exist Markov processes that fail it, so it must be earned (e.g. via the Feller property), not assumed.