how much time does brownian motion spends at 0? even though it spends lebesgue 0 amount of time there, this is not the end of the story. the local time for brownian motion let us go deeper into this question, creating an object that keeps track of how much time brownian motion spends at 0. once established, we will discuss applications of local time such as tanaka's formula, change of variables formula, engelbert-schimdt 0-1 law and reflected diffusions.