In the mathematical theory of stochastic processes, local time is a property of diffusion processes like Brownian motion that characterizes the time a particle has spent at a given level. Local time is very useful and often appears in various stochastic integration formulas if the integrand is not sufficiently smooth, for example in Tanaka's formula.
where is the diffusion process and is the Dirac delta function. It is a notion invented by P. Lévy. The basic idea is that is a (rescaled) measure of how much time has spent at up to time . It may be written as
which explains why it is called the local time of at .