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I know the link between the Fokker–Planck equation and SDE given by the Feynman-Kac theorem is as follow: $$d X_{t}=\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{t}$$ $$\frac{\partial}{\partial t} p(x, t)=-\frac{\partial}{\partial x}[\mu(x, t) p(x, t)]+\frac{\partial^{2}}{\partial x^{2}}[D(x, t) p(x, t)]$$ with $\sigma = \sqrt{2D}$.
I found a version of the proof, in Öttinger's 1996 book Stochastic Processes in Polymeric Fluids. However the Fokker–Planck equation is used with problems where the initial distribution is known, but the problem here is to know the distribution at previous times, which means that it is necessary to prove the Keynman–Kac forumula. I know the full proof of the Keynman–Kac foumula is complicated, but it may not be needed to prove the above conclusion. Can anyone provide some clues or reference for the part proof of Keynman–Kac formula? (By the way, how do you find a proof for a specific theorem.)

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  • $\begingroup$ for a text book you could look at H. Risken's "The Fokker-Planck Equation" $\endgroup$
    – Carlo Beenakker
    Commented May 14, 2021 at 21:56
  • $\begingroup$ It's a parabolic differential equation. Whatever the method you use to solve it you need to provide an initial condition $p(x,0)$. $\endgroup$
    – Daniel Castro
    Commented May 23, 2021 at 11:56

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