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I was going through Van Kampen's Stochastic Processes in Physics and Chemistry, and I was trying to solve the exercises from Chapter 8 about the Fokker Planck equation (just in case context could help anyone understand my question).

There, we have been asked to assume that some $P(y,t|y_0, t_0)$ does solve the Fokker Planck equation. We further assume that $t=t_0+\Delta t$ and that $\Delta t \rightarrow 0$. Then we have been asked to calculate the moments of $\Delta y$, which in turn is defined as $\Delta y \equiv y-y_0$, and thence calculate $\dfrac{<\Delta y>^j}{\Delta t}$. We have to show that $\dfrac{<\Delta y>}{\Delta t}=A(y_0)$, $\dfrac{<\Delta y>^2}{\Delta t}=B(y_0)$, and $\dfrac{<\Delta y>^j}{\Delta t}=0$ for $j \geq 3$.

Now, we have our Fokker Planck equation expressed as: $$ \frac{\partial P(y,t)}{\partial t}=-\frac{\partial}{\partial y}[A(y) P(y,t)]+\frac{1}{2}\frac{\partial^2}{\partial y^2}[B(y) P(y,t)] $$

This is how far I got: we can write down the definitions of our moments from first principles as (I will just write down the first moment of $\Delta y$).

$$ \frac{<\Delta y>}{\Delta t}=\int \frac{(y-y_0)}{\Delta t}P(y,t+\Delta t|y_0, t_0)dy $$ Now, given that $\Delta t$ is small we can write $$ \begin{align} P(y,t_0+\Delta t|y_0, t_0)&=P(y,t_0|y_0, t_0)+\Delta t \frac{\partial}{\partial t}P(y,t|y_0, t_0)\vert_{y_0, t_0}\\ &=\delta(y-y_0)+\Delta t \frac{\partial}{\partial t}P(y,t|y_0, t_0)\vert_{y_0, t_0} \end{align} $$ When we plug this back into the integral, the delta function term won't contribute anything and we can work only with the second term:

$$ \frac{<\Delta y>}{\Delta t}=\int \frac{(y-y_0)}{\Delta t}\Delta t \frac{\partial}{\partial t}P(y,t|y_0, t_0)\vert_{y_0, t_0}dy $$

We can cancel out the $\Delta t$ and then use the fact that our $P$ solves the FPE to get:

$$ \frac{<\Delta y>}{\Delta t}=\int (y-y_0) \Big[-\frac{\partial}{\partial y}[A(y) P(y,t|y_0, t_0)]+\frac{1}{2}\frac{\partial^2}{\partial y^2}[B(y) P(y,t|y_0, t_0)]\Big]\Big\vert_{y_0, t_0}dy $$

I am not sure how to proceed from here. My first instinct is to try integrating it by parts, but I don't know the limits of integration, so I am unaware of how to proceed. Additionally, I do have a feeling there is a more elegant way to arrive at this result because these moments do describe convection and diffusion, and $A$ and $B$ are the convection and diffusion terms (am not sure if I am putting the cart before the horse here, but it does seem quite intuitive).

Any help would be appreciated!

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The procedure that you propose is right! I am actually not aware of any other strategy.

Try the integration by parts bearing in mind that you know the distribution $P(y,t_0|y_0,t_0)=\delta(y-y_0)$.

If the boundary conditions are not given, then it is usually assumed that the probability and the probability flux are zero at $\pm \infty$. This is a way to obtain normalizable probabilities.

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  • $\begingroup$ I think I got it! Also, do you mean that $P(y,t|y_0, t_0)=\delta(y-y_0)$ only at $t=t_0$ (because $P$ can definitely have a non-trivial dependency on $t$). Because this condition allows me to in some sense 'collect' whichever moment I want from the FPE. $\endgroup$
    – VBtheHun
    Commented May 26, 2022 at 9:52
  • $\begingroup$ Exactly, It was a typo. Edited. $\endgroup$
    – Javi
    Commented May 26, 2022 at 10:59

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