In the Wikipedia article about Langevin equations they say that
The force $\eta(t)$ has a Gaussian probability distribution with correlation function $$\langle \eta(t) \eta(t')\rangle = 2\lambda k_B T \delta(t-t').$$
My question is, where does this formula come from? My understanding is that the autocorrelation is given by $$\begin{aligned} R_{\eta\eta}(t,t')&=E[\eta(t)\eta(t')] \\ &=\begin{cases} \sigma^2_\eta, & t = t', \\ 0, & t\ne t', \end{cases} \end{aligned}$$ where $\sigma_\eta^2$ is the variance of $\eta(t)$. Why is the formula above for the Langevin equation infinite at $t=t'$? Could you show me where that equation comes from and how $2\lambda k_B T$ relates to the variance of $\eta(t)$?
The noise term, $\eta$, is a Gaussian white noise. That is, it is a stationary mean-zero Gaussian process with flat spectral density $S_{\eta}\equiv p$ ($p$ is the "power level" of $\eta$). The autocorrelation function of $\eta$ is the inverse Fourier transform of $S_{\eta}$, i.e. $$ R_{\eta}(s)=\mathcal{F}^{-1}(S_{\eta})=p\delta(s), $$ where $\delta$ is the Dirac delta function. Specifically, the variance of $\eta(t)$ is infinite because otherwise the power spectrum would be a null function.