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?

Got something from an answer below? – Did Jul 04 '13 at 14:04 — Did, Jul 04 '13 at 14:04

score 1 · Answer 1 · answered Mar 22 '13 at 17:28

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.

score 0 · Answer 2 · answered Mar 22 '13 at 19:36

0

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

answered Mar 22 '13 at 19:36

roger

2,986

why we cannot properly build uncountable many independent random variables ? – kerzol Jul 27 '14 at 10:14
This is addressed in https://math.stackexchange.com/questions/1549807/showing-that-there-do-not-exist-uncountably-many-independent-non-constant-rando – gigalord Jan 09 '20 at 20:54

demorge · Answer 3 · 2013-03-22T18:30:09.450

-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.

edited Mar 22 '13 at 18:30

answered Mar 22 '13 at 17:28

demorge

293

Stochastic process with delta correlation in time

3 Answers3