I am looking at this paper Smoluchowski diffusion equation for active Brownian swimmers , in which they describe using something called Novikov's theorem to take correlations between a random force and a delta function that describes density. I have tried reading the references they provide, but the math is just beyond me. Has anyone encountered this or do they have a clear explanation?
From the paper:
Given a stochastic brownian processes $\beta(t)$, described by a vanishing average $<\beta(t)> = 0$ and correlation $<\beta(t)\beta(s)> =2\gamma\delta(t-s)$
The functional of that process $F[\beta(t)]$ has the property
$$ <F[\beta(t)]\beta(t)> = \gamma<\frac{\tilde{\delta}F[\beta(t)]}{\tilde{\delta}\beta(t)}> $$ Where the right hand side is the variational derivative.
They then use that theorem to get this... (moving from equations 2 to 3)
$$ -\nabla\cdot<\xi(t)\delta(x-x(t))\delta(\varphi-\varphi(t)> = D_B\nabla^2P(x,\varphi,t) $$
If anyone can explain that last step to me I would really appreciate it. I imagine somehow it is stating that $$\frac{\tilde{\delta} \delta(x-x(t)) *\delta(\varphi-\varphi(t))}{\tilde{\delta}\xi(t)} = D_B\nabla P(x,\varphi,t)$$
But I do not understand how that would occur.
Thank you!