Fokker Planck and SDE I have the following Fokker-Planck equation in spherical coordinates $(\theta,\phi)$:

$$ \partial f/ \partial t= D   \cot\theta \quad \partial f/\partial \theta + \quad 1/\sin^2\theta \quad \partial^2 f/\partial \phi^2 - \quad A[\sin\theta \partial f/\partial\theta +2 \cos\theta f] \tag{1}$$

where $D$ and $A$ are constants. I want to write it in stochastic differential equation form. I have no idea about the stochastic differential equation but I am reading from Gardiner book ''Handbook of Stochastic Methods'' 2 edition.
In the book, the Fokker-Planck equation in one dimension:

$$ \partial f/\partial t= - \partial /\partial x [A(x,t) f(x,t)]+1/2 \partial^2/\partial x^2 [B(x,t) f(x,t)] \tag{2}$$  

It can be written in Ito SDE form-

$$ dx(t)=A(x,t) dt +\sqrt{B(x,t)}\; dW(t) \tag{3}$$

Note that Fokker-Planck equation needs to be in certain format as in (2) so that you can write its equivalent SDE. Any help will be appreciated.

The relation between the Fokker-Planck equation and the associated SDE has been investigated by Figalli (2008) and is known as the (stochastic analogue of the) superposition principle.
Michael Röckner and colleagues extended the superposition principle to many different situations, e.g. for McKean-Vlasov SDEs or non-local Fokker-Planck equations.
