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The occupation time of a measurable set $A$ through time $t$ is ${\int }_0^t{1}_{\{B_s\in A\}}ds$. The occupation-times formula expresses such integrals through local time of a semimartingale. Recurrence and the Strong Markov property imply that one-dimensional Brownian motion spends an unbounded total time in every nonempty open interval.

For a continuous semimartingale $X$ with local time of a semimartingale $L_t^a(X)$,for every nonnegative measurable function $f$.

One-dimensional reflected Brownian motion on $[0,\infty )$ has the same law as $|B_t|$. The Tanaka formula represents it as a Brownian motion plus a nondecreasing local time of a semimartingale that grows only at zero.

For a continuous semimartingale $X$,where $L^0(X)$ is the local time of a semimartingale at zero.