Derivation of Kramer's equation For the derivation of Kramer's equation we use the multivariable Fokker-Planck equation: 
$$\frac{\partial P}{\partial t} = \frac{\partial (P A_{1})}{\partial x} + \frac{\partial (P A_{2})}{\partial v} +\frac{1}{2} \frac{\partial^{2} (P B_{11})}{\partial x^2} + \frac{\partial^{2} (P B_{12})}{\partial v \partial x} + \frac{1}{2} \frac{\partial^{2} (P B_{22})}{\partial v^2}$$
 where, $$ A_{1} = \frac{<\Delta{x}>}{\Delta{t}} , A_{1} = \frac{<\Delta{v}>}{\Delta{t}} ,$$$$ B_{11} = \frac{<(\Delta{x})^2>}{\Delta{t}} ,B_{12} = \frac{<\Delta{x}\Delta{v}>}{\Delta{t}} ,B_{22} = \frac{<(\Delta{v})^2>}{\Delta{t}} $$
Final form was derived using the following conditions:
$$ (\Delta{x})^2 \rightarrow 0$$ $$ (\Delta{x} \Delta{v}) \rightarrow 0$$
I couldn't find the reason behind these conditions. Can someone please explain it to me.
 A: $B_{12}$ and $B_{11}$ are proportional to $\Delta(t)^m$,  so when we take small $\Delta(t)$ limit and they vanish. 
But in case of $<(\Delta v)^2>$, the term proportional to $\Delta t$ so $B_{22}$ survives. 
because $<dW\cdot dw> = \int\int dt'dt <\xi(t)\xi(t')> = (costant)\cdot\int\int dt'dt \delta(t-t') = (constant)\int dt = (constant)\cdot\Delta t$ 
( $W(t)$ is also called Wigner process. )
Here, i use Langevin equation $m \dot{v} = f(x) -\gamma v + \xi (t) $ where $\xi $ is uncorrelated Gaussian noise $<\xi(t)> = 0$, $<\xi(t)\xi(t')>=(constant)\delta(t-t')$. 
 And this constant can be determined by Einstein relation.
Anyway from below relations you can derive Kramer's equation
$d x = v dt $ 
$d v = f(x)dt-\gamma v dt + \xi dt =  f(x)dt-\gamma v dt + dW(t) $
Another way is that if the stochastic process satisfies below conditions then you can prove higher terms are gone away 
1) $lim_{\delta t\rightarrow 0} \frac{1}{\delta t} \int _{x-z<\eeta}dx(x-z)p(x,t+\delta t|z,t) = A(z,t) + \bigo{\eeta} $
2) $lim_{\delta t\rightarrow 0} \frac{1}{\delta t} \int _{x-z<\eeta}dx(x-z)p(x,t+\delta t|z,t) = B(z,t) + \bigo{\eeta}$
Detail derivation is in Handbook of stochastic method chater 3.4 
