2
$\begingroup$

I am trying to learn stochastic calculus and when they talk about the Langevin equation they say that the correlation of the gaussian white noise (which i believe is the covariance between two random variables in the stochastic process) has a "$\delta$ correlation in time":

$\langle\xi(t)\xi(\tau)\rangle=\delta(t-\tau)$

Where $\delta$ is "the delta function". Now I wonder which one they mean, is it the generalised function (does that mean the correlation is $\infty$?) or is it the indicator function? Or something entirely different?

$\endgroup$
  • $\begingroup$ Got something from an answer below? $\endgroup$ – Did Jul 4 '13 at 14:04
1
$\begingroup$

I think that when something like this is said, it is meant, that the processe $\xi(t)$ is correlated only at one single moment of time: that's the meaning of delta-function term. :) So the autocorrelation function vanishes in other moments. In some way it is the indicator function (as your have said) of the moment when it is not zero.

$\endgroup$
0
$\begingroup$

There is no such thing as a continuous white noise. You can not properly build uncountably many independent random variables.

$\endgroup$
-1
$\begingroup$

The sharp braces indicate the ensemble average, using the Kronecker delta means its only fully positively correlated with itself when $t = \tau$. For any $t \neq \tau$ it is uncorrelated.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.