Local time (mathematics)

In the mathematical theory of , local time is a stochastic process associated with diffusion processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields.

For a real valued diffusion process ${\displaystyle (B_{s})_{s\geq 0}}$, the local time of ${\displaystyle B}$ at the point ${\displaystyle x}$ is the stochastic process

where ${\displaystyle \delta }$ is the Dirac delta function. It is a notion invented by Paul Lévy. The basic idea is that ${\displaystyle L^{x}(t)}$ is a (rescaled) measure of how much time ${\displaystyle B_{s}}$ has spent at ${\displaystyle x}$ up to time ${\displaystyle t}$. It may be written as

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