# Brownian noise correlations

In a (physics) paper I have seen a stochastic differential equation $$\frac{\mathrm{d} x}{\mathrm{d} t} = f(x, t) + W(x)$$ where $$W$$ is described as Brownian noise written with $$\left\langle W(x) \right\rangle = 0$$ and $$\left\langle W(x) W(x^\prime) \right\rangle = |x-x^\prime|$$. By taking a derivative with respect to $$x$$ the author arrives to $$\frac{\mathrm{d} v}{\mathrm{d} x} = \frac{\mathrm{d} f}{\mathrm{d} x} + w(x)$$ where $$v = \mathrm{d} x / \mathrm{d} t$$ and the noise term $$w(x) = \frac{\mathrm{d} W}{\mathrm{d} x}$$ has $$\left\langle w(x) \right\rangle = 0$$ and $$\left\langle w(x) w(x^\prime) \right\rangle = \delta(x-x^\prime)$$.

This makes sense to me as white noise ($$w$$ here) can be understood as the formal derivative of Brownian motion and the delta-correlation seems right. Although I can't figure out how one gets this $$|x-x^\prime|$$ correlation out of Brownian noise or transforms between these two.

Also in an another paper a similar equation is written with $$\left\langle \mathrm{d} W \right\rangle = 0$$ and $$\left\langle | \mathrm{d} W |^2 \right\rangle = \mathrm{d} x$$ which perplexes me even further. How should I properly deal with these?