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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.

- $ell(t,x)=int\_0^t\; delta(x-b(s)),ds$

where $b(s)$ is the diffusion process and $delta$ is the Dirac delta function. It is a notion invented by P. Lévy. The basic idea is that $ell(t,x)$ is a (rescaled) measure of how much time $b(s)$ has spent at $x$ up to time $t$. It may be written as

- $ell(t,x)=lim\_\{epsilondownarrow\; 0\}\; frac\{1\}\{2epsilon\}\; int\_0^t\; 1\{\; x-\; epsilon\; <\; b(s)\; <\; x+epsilon\; \}\; ds,$

which explains why it is called the local time of $b$ at $x$.

- Tanaka's formula
- Brownian motion
- Red noise, also known as brown noise (Martin Gardner proposed this name for sound generated with random intervals. It is a pun on Brownian motion and white noise.)
- Diffusion equation

- K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, 1990, Birkhäuser, ISBN 978-0817633868 .

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Last updated on Tuesday April 22, 2008 at 15:55:18 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday April 22, 2008 at 15:55:18 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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