Stochastic process with delta correlation in time

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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