I was trying to proof Milstein Algorithm so I read the book "Stochastic Numerical Methods_ An Introduction for Students and Scientists" of Toral. Here, they show all the proof in one dimension and many dimensions but just considerer white noise in "time", not in space.
I was also reading the book "Noise in Spatially Extended Systems" of Garcia Ojalvo and they do considerer white noise in space and time. However, there is an approximation that I can't proof. The image is attached in the following link:
Photo of Spatial white-noise limit
I think that this expression can be get if the following approximation is true:
$$\lim_{x' \rightarrow x} \delta(x'-x) = \frac{1}{(x'-x)}$$
where $\delta$ is the Dirac Delta function
There is a way to prove it? or Maybe it is not the path to prove space white-noise limit