the reflection principle
The reflection principle is the elegant symmetry trick that lets you compute the distribution of the running maximum of Brownian motion, and far more, by folding paths across a level. The picture: each Brownian path that exceeds a level a at some time before t and ends below a can be paired with a reflected twin that ends above a, and the two are EQUALLY likely. Counting paths through this mirror converts a hard question about a maximum over an interval into an easy question about the position at a single time.
Concretely, fix a > 0 and let tau = inf{ t : B_t = a }. The event {max_(u<=t) B_u >= a} is the same as {tau <= t}. On this event, the strong Markov property says the post-tau path B_(tau+s) - a is a fresh Brownian motion, symmetric about 0. Reflecting it (flipping its sign after tau) produces a path of the same law that ends at 2a - B_t instead of B_t. Hence P(tau <= t, B_t < a) = P(tau <= t, B_t > a) = P(B_t > a) (the last because reaching above a forces tau <= t). Adding the two equal pieces gives P(max_(u<=t) B_u >= a) = 2 P(B_t >= a) = 2 P(B_t > a). The same reflection at tau gives the joint law of (max_(u<=t) B_u, B_t).
Why it matters: the reflection principle is the source of the maximum's distribution, the first-passage (one-sided stable) density, the arcsine laws, and ballot-type problems; it also underlies the construction of reflected and absorbed Brownian motions and the method of images for the heat equation on a half-line. The crucial hypotheses are the strong Markov property AND the symmetry of Brownian increments — reflection works because the fresh post-tau motion is symmetric. For a Brownian motion with nonzero drift the naive reflection fails, and one must reflect AND reweight by a Girsanov factor; do not apply the symmetric version to a drifted process.
For standard Brownian motion, P(max_(u<=1) B_u >= 2) = 2 P(B_1 >= 2) = 2 (1 - Phi(2)) approximately 0.0455, where Phi is the standard normal CDF. So the maximum over [0,1] exceeds 2 about twice as often as the endpoint B_1 alone does.
Folding paths at the first hit of a converts a maximum-over-an-interval question into a position-at-a-time question.
The factor 2 comes from the symmetry of the increments after the hit; for Brownian motion WITH drift the symmetric reflection is wrong and must be corrected by a Girsanov change of measure.