Imagine that I have a stochastic process that I hypothesize evolves according to the 1-Dimensional Fokker-Planck equation:
Let $x$ denote the physical property.
Let $t$ denote time, $t_0$ denote a specific time, and $t_1 > t_0$ denote a specific time after $t_0$.
Let $W(x,t)$ denote the probability density function of $x$ at time $t$.
Then, in general, the physical process evolves according to $\frac{\partial W(x,t)}{\partial t} = W(x,t) \left[ -\frac{\partial}{\partial x}S(x,t)+\frac{\partial^2}{\partial x^2}M(x,t) \right]$.
Assume $S$ is constant over $t$ and $M$ is constant over both $x$ and $t$. We can simplify to $\frac{\partial W(x,t)}{\partial t} = -\frac{\partial}{\partial x}(S(x)W(x,t))+\frac{\partial^2}{\partial x^2}(MW(x,t)) $
Further, assume $S(x)$ is quadratic, i.e., $S(x) = \frac{\gamma}{2}x^2 + \beta x + \alpha$
My question is: If we are given two distributions,$W(x,t_0)$ and $W(x,t_1)$, can we solve for the most likely coefficients $\gamma$,$\beta$, $\alpha$, and $M$, if we assume that $W(x,t_1)$ was reached by the evolution of the Fokker-Planck equation from the initial condition $W(x,t_0)$?
If this proves to be impossible because there are too many unknown varaibles, we can make further simplifying assumptions by first assuming $\alpha =0$ , and then assuming $\gamma = 0$.