Let
$$X_t= X_0 + \int_0^t \mu(X_s,s) \, ds + \sigma B_t$$
where $B_t$ is a one dimensional standard Brownian motion and $\sigma >0$.
The Fokker-Planck equation describes the evolution of the density $p(x,t)$ of $X_t$:
$$\frac{\partial}{\partial t} p(x, t)=-\frac{\partial}{\partial x}[\mu(x, t) p(x, t)]+\frac{\sigma^2}{2}\frac{\partial^{2}}{\partial x^{2}}[ p(x, t)]$$
Usually, the drift $\mu$ is given in the problem. Here, I am actually looking for the drift, not $p(x,t)$. What I can tell you about $p(x,t)$ is that it has $p(x,0)=p(x,1)$, and let's say for simplicity that $p(x,0)=1_{[0,1]}(x)$ (the density of a uniform measure on $[0,1]$). For now, I do not care about what happens with the density between times $0$ and $1$, it can be any continuous density that satisfies the constraint (as long as it is not constant in time).
How can I find a drift such that $p(x,0)=p(x,1)$ ? Is there at least one ? If so, how many are there ? Can we compute them ? Can we find the density of $\mu(X_t,t)$ in terms of $p(x,t)$ ? Or at least its moments in terms of the moments of $X_t$ ?