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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?

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