In mathematics, particularly in the field of topology and analysis, "local time" refers to a concept that describes the time evolution of a stochastic process, especially in the context of Brownian motion and other random processes. Local time helps to quantify how often a process visits a particular state or value over time. For instance, in the context of Brownian motion, local time can be viewed as a way to record the "amount of time" the Brownian motion spends at a particular level.
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Local time records how much a stochastic process visits a level. Its normalization depends on the process: when the state space is a discrete space, it is occupation time at a state, while local time of a semimartingale is an occupation density weighted by quadratic variation. Brownian local time is a special case.