# Confusion about the covariance of the Wiener process

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?

• – user140541
Apr 12, 2022 at 11:49
• A way of formalizing the idea of "white noise" in continuous time is to use the theoretical apparatus developed by Hida. I advice you to give a look to the book link.springer.com/book/10.1007/978-94-017-3680-0 Apr 21, 2022 at 8:35
• thanks @Chaos I will have a look. Apr 21, 2022 at 9:52