Local time (mathematics)

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

is the stochastic process which is informally defined by where

is the Dirac delta function and

It is a notion invented by Paul Lévy.

is an (appropriately rescaled and time-parametrized) measure of how much time

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

is a Brownian motion), the term

, which explains why it is called the local time of

For a discrete state-space process

, the local time can be expressed more simply as[1] Tanaka's formula also provides a definition of local time for an arbitrary continuous semimartingale

[2] A more general form was proven independently by Meyer[3] and Wang;[4] the formula extends Itô's lemma for twice differentiable functions to a more general class of functions.

is absolutely continuous with derivative

has a modification which is a.s. Hölder continuous in

Tanaka's formula provides the explicit Doob–Meyer decomposition for the one-dimensional reflecting Brownian motion,

associated to a stochastic process on a space

is a well studied topic in the area of random fields.

Ray–Knight type theorems relate the field Lt to an associated Gaussian process.

In general Ray–Knight type theorems of the first kind consider the field Lt at a hitting time of the underlying process, whilst theorems of the second kind are in terms of a stopping time at which the field of local times first exceeds a given value.

Let (Bt)t ≥ 0 be a one-dimensional Brownian motion started from B0 = a > 0, and (Wt)t≥0 be a standard two-dimensional Brownian motion started from W0 = 0 ∈ R2.

Define the stopping time at which B first hits the origin,

Ray[6] and Knight[7] (independently) showed that where (Lt)t ≥ 0 is the field of local times of (Bt)t ≥ 0, and equality is in distribution on C[0, a].

Let (Bt)t ≥ 0 be a standard one-dimensional Brownian motion B0 = 0 ∈ R, and let (Lt)t ≥ 0 be the associated field of local times.

Let Ta be the first time at which the local time at zero exceeds a > 0 Let (Wt)t ≥ 0 be an independent one-dimensional Brownian motion started from W0 = 0, then[8] Equivalently, the process

(which is a process in the spatial variable

) is equal in distribution to the square of a 0-dimensional Bessel process started at

Results of Ray–Knight type for more general stochastic processes have been intensively studied, and analogue statements of both (1) and (2) are known for strongly symmetric Markov processes.