I often see the following stochastic function $\Gamma(x)$ defined based on only these requirements:
$$\langle\Gamma(x)\rangle=0$$ $$\langle\Gamma(x)\Gamma(x')\rangle=\delta(x-x')$$ where $\delta$ is the Dirac delta function.
My question is simply, what is $\Gamma$? Or, what is the class of functions that satisfy these conditions?
I suspect, based on the context I see this function in (Fokker-Planck / Langevin type equations), that $\Gamma(x) \sim \mathcal{N}(x)$, i.e., gamma has the same form as a normal distribution.
It seems that $\langle f(x)f(x')\rangle$ is related to the auto-correlation of $f$. Therefore, the condition $\langle f(x)f(x')\rangle = \delta(x-x')$ is saying that the auto-correlation is zero everywhere except equality.
As @Tobsn pointed out in the comments, this is a condition of white noise. We can look to this post or this Wikipedia entry for more clarity. In particular, we find the relationship between $\Gamma$ and $\mathcal{N}$: their characteristic functions are similar in form.