Confusion about the covariance of the Wiener process

Question

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?

Answer 1

As mentioned in What is meant by a continuous-time white noise process?, Importance of white noise and Brownian motion and What is "white noise" and how is it related to the Brownian motion?, we can have define the White-noise process $X$ as a distribution over differentiable functions in terms of Brownian motion

\begin{equation} (X, f) = -(B, f') = -\int_0^\infty B(t) f'(t) dt. \end{equation} Or alternatively using the Itô integral as a distribution over $L^{2}$-functions \begin{equation} (X, f) = \int_0^\infty f(t) dB_t, \end{equation} (this is the most common function space to define white-noise over).

From here we indeed observe that by taking $f_{i}(t)=\frac{1}{\epsilon}1_{B_{\epsilon}(x_{i})}(t)$ we have by Itô-isometry \begin{align} E[(X(x_{1})(X(x_{2})]\approx& E[(X, f_{1})(X, f_{2})] = E\int_0^\infty f_{1}(t)f_{2}(s) dB_t dB_s\\ =&\int_0^\infty f_{1}(t)f_{2}(t)dt=c\frac{1}{\epsilon}1_{|x_{2}-x_{1}|<\epsilon}\approx \delta(|x_{2}-x_{1}|). \end{align}