In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various stochastic integration 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 continuous real-valued semimartingale ${\displaystyle (B_{s})_{s\geq 0}}$, the local time of ${\displaystyle B}$ at the point ${\displaystyle x}$ is the stochastic process which is informally defined by

where ${\displaystyle \delta }$ is the Dirac delta function and ${\displaystyle [B]}$ is the quadratic variation. It is a notion invented by Paul Lévy. The basic idea is that ${\displaystyle L^{x}(t)}$ is an (appropriately rescaled and time-parametrized) measure of how much time ${\displaystyle B_{s}}$ has spent at ${\displaystyle x}$ up to time ${\displaystyle t}$. More rigorously, it may be written as the almost sure limit

which may be shown to always exist. Note that in the special case of Brownian motion (or more generally a real-valued diffusion of the form ${\displaystyle dB=b(t,B)\,dt+dW}$ where ${\displaystyle W}$ is a Brownian motion), the term ${\displaystyle d[B]_{s}}$ simply reduces to ${\displaystyle ds}$, which explains why it is called the local time of ${\displaystyle B}$ at ${\displaystyle x}$. For a discrete state-space process ${\displaystyle (X_{s})_{s\geq 0}}$, the local time can be expressed more simply as

Tanaka's formula also provides a definition of local time for an arbitrary continuous semimartingale ${\displaystyle (X_{s})_{s\geq 0}}$ on ${\displaystyle \mathbb {R} :}$

A more general form was proven independently by Meyer and Wang; the formula extends Itô's lemma for twice differentiable functions to a more general class of functions. If ${\displaystyle F:\mathbb {R} \rightarrow \mathbb {R} }$ is absolutely continuous with derivative ${\displaystyle F',}$ which is of bounded variation, then

where ${\displaystyle F'_{-}}$ is the left derivative.

If ${\displaystyle X}$ is a Brownian motion, then for any ${\displaystyle \alpha \in (0,1/2)}$ the field of local times ${\displaystyle L=(L^{x}(t))_{x\in \mathbb {R} ,t\geq 0}}$ has a modification which is a.s. Hölder continuous in ${\displaystyle x}$ with exponent ${\displaystyle \alpha }$, uniformly for bounded ${\displaystyle x}$ and ${\displaystyle t}$. In general, ${\displaystyle L}$ has a modification that is a.s. continuous in ${\displaystyle t}$ and càdlàg in ${\displaystyle x}$.