# Converting Fokker-Planck equation to some form

Fokker-Planck equation (one-dimension) is:

$$\tag{0} \frac{\partial}{\partial t}f(x,t) = -\frac{\partial}{\partial x}\left[\mu(x,t)f(x,t)\right] + \frac{\partial^2}{\partial x^2}\left[ D(x,t)f(x,t)\right].$$

then, how does one convert to

$$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\partial x}+D_2 \left( x,t\right) \frac{\partial^2}{\partial x^2}\right] \delta\left( x^{\prime}-x\right) \right) f\left( x,t\right).\qquad$$?

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How do you know (1) holds? Which related results do you know? What did you try? – Did Mar 1 at 9:09
I tried using ito lemma. – uvr Mar 1 at 9:21
Could you be more specific? – Did Mar 1 at 9:25
dividing dX by dt...+ito's lemma – uvr Mar 1 at 9:27
?? OK, obviously you are not even trying to answer the questions in my first comment. So be it. – Did Mar 1 at 9:30