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So I am a physicist, and in general I deal with so-called Langevin equations of the type \begin{equation} \frac{dx}{dt} = f(x,t) +g(x,t)\xi(t), \end{equation} where $\xi(t)$ is what we call a "random Langevin force". In the case of white noise, this force is delta-correlated in time, such that \begin{equation} \mathbf{E}\left[\xi(t)\xi(t')\right] = \delta(t-t'). \end{equation} The Langevin equation can also be expressed in terms of the Ito equation \begin{equation} dx = f(x,t)dt+g(x,t)dW(t), \end{equation} where the Wiener process $dW(t)=\xi(t)dt$. Basically this means that \begin{equation} \xi(t) = \frac{dW(t)}{dt}, \end{equation} even though technically the Wiener process is not differentiable. This is where things get confusing for me. I know that $\mathbf{E}[dW(t)] = 0$ and Var$[dW(t)] = dt$, but how exactly is it that $\xi(t)$ has delta autocorrelation? Is there a mathematical way to get to this result? \begin{equation} \mathbf{E}\left[\xi(t)\xi(t')\right] = \mathbf{E}\left[\frac{dW(t)}{dt}\frac{dW(t')}{dt'}\right] = \delta(t-t')\ ? \end{equation} I have looked everywhere in the literature and I can't see how this is obtained, it seems to be a detail that is glossed over all the time, especially in physics.

If $\xi(t)$ is truly represented as the rate of change in the Wiener process $W(t)$, then surely there must be a mathematical way to derive the result $\mathbf{E}\left[\xi(t)\xi(t')\right] = \delta(t-t')$ from the properties of $W(t)$ alone right?

OAN
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  • The autocorrelation of BM $W(t)$ is $\operatorname{min}(s,t)$, which differentiated with respect to $s$ and $t$ yields $\delta(s-t)$. – Tobsn Jun 21 '21 at 10:18
  • @Tobsn do you have any links/books where that is shown? I am actually not familiar with that – OAN Jun 21 '21 at 10:43
  • e.g. https://link.springer.com/content/pdf/bbm%3A978-3-540-36852-6%2F1.pdfv – Tobsn Jun 21 '21 at 12:03
  • @Tobsn Awesome, thanks, what book is this from? I might need to cite this on my work – OAN Jun 22 '21 at 02:17

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A formal connection with white noise is described in Le-Gall "Brownian Motion, Martingales, and Stochastic Calculus". See formal descriptions here What is "white noise" and how is it related to the Brownian motion? too that describe their unique connection.

If we use the distributional definition of white noise $\xi$

$$(\xi,\phi)=-\int B_{t}\phi'(t)dt,$$

then the covariance of white at two different times $S,T$ can be approximated by a normalized:

$$E\left[(\xi,1_{[S-\epsilon,S+\epsilon]})E(\xi,1_{[T-\epsilon,T+\epsilon]})\right]=E\left[\iint 1_{[S-\epsilon,S+\epsilon]]}(r)1_{[T-\epsilon,T+\epsilon]]}(t) dB_{r}dB_{t}\right]=\int 1_{[S-\epsilon,S+\epsilon]]\cap [T-\epsilon,T+\epsilon]]}(r)dr,$$

and so in the limit after normalizing by $\frac{1}{\epsilon}$, we get $0$ or $+\infty$ depending on whether $S\neq T$ or $S= T$. (One has to mollify with compactly supported smooth functions $\phi_{\epsilon,S}(r),\phi_{\epsilon,T}(r)$).

Thomas Kojar
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