This is inspired by the following question. Let $X_t$ be an Ito diffusion on the interval $t\in [0,1]$: $$ \mathrm dX_t = a(X_t)\mathrm dt+ b(X_t)\mathrm dW_t $$ where say $a,b$ are Lipschitz continuous functions. Let us denote by $\mathcal P(\Bbb R)$ the class of all Borel measures on $\Bbb R$.
Let $\alpha\in \mathcal P(\Bbb R)$. Is it possible to characterize the class $\mathcal P_\alpha$ of all possible distributions of $X_1$ given the fact that $X_0\sim \alpha$?
Given $\beta\in \mathcal P_\alpha$, is it possible to choose $a$ and $b$ in some constructive way?