Brownian local time
Brownian motion spends zero Lebesgue time at any single point (the time at any exact level is a measure-zero set), so 'how long does B stay at level a?' has the trivial answer zero. Local time L_t^a is the right renormalised answer: it measures the AMOUNT of time, suitably densified, that the path spends near a up to time t. It turns the singular question into a smooth, increasing clock that ticks only while B is exactly at a.
Formally, local time is the occupation density: define it so that, for every bounded measurable f, integral over [0,t] of f(B_s) ds = integral over R of f(a) L_t^a da. Equivalently L_t^a = lim_(epsilon -> 0) (1 / (2 epsilon)) Leb{ s <= t : |B_s - a| < epsilon }, the time spent in an epsilon-band around a, normalised by the bandwidth. A theorem (Trotter) says one can choose a version of L_t^a that is jointly continuous in (t, a). As a function of t, L_t^a is nondecreasing and increases only on the (measure-zero) set { s : B_s = a }, so it is a singular clock supported on the zero set of B - a. The cleanest construction is via Tanaka's formula: |B_t - a| = |B_0 - a| + integral over [0,t] of sgn(B_s - a) dB_s + L_t^a, which exhibits L_t^a as the increasing process making |B - a| a semimartingale — local time is exactly the 'extra' arising because |x - a| is not smooth at a.
Why it matters: local time is the bridge between the path and its zero set, the key to excursion theory (excursions are indexed by local time), the Ray-Knight theorems (which describe a -> L_t^a as a Bessel-squared process), reflected and skew Brownian motions (built using local time at 0), and additive-functional and PDE problems with a singular source at a level. The honest subtleties: L_t^a is NOT a time in the ordinary sense — it has its own scale, and Levy's theorem shows the process L_t^0 has the SAME law as the running maximum M_t, even though one grows on the zero set and the other tracks record highs. And local time is defined only up to the choice of a continuous version; the symmetric vs one-sided conventions differ by a factor.
Levy's theorem says the pair (M_t - B_t, L_t^0) has the same law as (|B_t|, M_t): the local time at zero accumulates exactly like the running maximum. So you can read the local-time clock off the maximum of a reflected path — they are different mechanisms with the same statistics.
Local time at a is the occupation density that increases only while B sits exactly at level a.
Local time is not elapsed clock time but a renormalised occupation density supported on the zero set; conventions (symmetric vs right-sided) differ by a factor of 2, and only the jointly continuous version is canonical.