When studying the Wiener process, I learned that the variance of this process is $Var(W_t) = t$, (which can be proven by calculating the quadratic variation) and furthermore that $\mathbb{E}W_tW_s = \min (t,s)$. However, looking into the Physics literature, one often comes across Langevin equations such as

$$dX(t) = u dt + v \xi(t) dt,$$

where $\xi(t)$ is said to be a Gaussian white noise with $\langle \xi(t)\rangle = 0 $ and $\langle \xi(t)\xi(t') \rangle = \delta( t-t')$. See for example [1]. Now what I expect is that $\xi(t)$ is actually shorthand for $\frac{dW_t}{dt}$ which of course is not well-defined but one could consider the ratio of increments $\Delta W_t /\Delta t$. However, even if this explanation were true, I still don't really know where the covariance $\langle\xi(t)\xi(t')\rangle = \delta (t-t')$ comes from. So my question is, what precisely is $\xi(t)$ and where does this expression for the covariance come from?



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