Let $X$ be the solution to (real-valued) stochastic differential equation :
$$dX_t = b(t,X_t)dt + a(t,X_t)dW_t, \quad \forall t\ge 0.$$
Let $\Delta t>0$ be given. Under suitable conditions (on $b,a, X_0$), do we have a PDE characterisation for the joint distribution of $(X_t, X_{t+\Delta t})$? More precisely, let $p(t,\cdot)$ be the density function of their joint distribution (if existing), can we expect a PDE of Fokker-Planck type for $p(t,x,y)$?
