 The basic idea is that formula_7 is an (appropriately rescaled and time-parametrized) measure of how much time formula_8 has spent at formula_3 up to time formula_10.More rigorously, it may be written as the almost sure limit

which may always be shown to exist.Note that in the special case of Brownian motion (or more generally a real-valued diffusion of the form formula_12 where formula_13 is a Brownian motion), the term formula_14 simply reduces to formula_15, which explains why it is called the local time of formula_2 at formula_3.