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The first equation, $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ is actually a shorthand notation for its correct full form, which is $$\underset{\epsilon\rightarrow0^+}{lim}\int_{-\infty}^\infty\frac{f(x)}{x-x_0+i\epsilon}\,dx=P\int_{-\infty}^\infty\frac{f(x)}{x-x_0}\,dx-i\pi f(x_0)$$ and is valid for functions which are analytic in the ...
Going from equation (0) to (1) is basically by writing the adjoint of the linear operator in the Fokker Planck equation (with a standard $L^2$ inner product). So the first equation is basically the following statements-  \dfrac{\partial f(x,t)}{\partial t}=\mathcal{L}_xf\text{ where }\mathcal{L}_x\equiv-\dfrac{\partial}{\partial ...