Given a two-dimensional Ito stochastic equations \begin{align} \mathrm{d} x &= \mu_1(x,y) \mathrm{d} t + \sigma_{11}(x,y) \mathrm{d} W_1 + \sigma_{12}(x,y) \mathrm{d} W_2, \\ \mathrm{d} y &= \mu_2(x,y) \mathrm{d} t + \sigma_{21}(x,y) \mathrm{d} W_1 + \sigma_{22}(x,y) \mathrm{d} W_2, \end{align} would it be possible to derive two Fokker-Planck equations for marginal probability density functions $p(x)$ and $p(y)$, instead of the joint probability $p(x,y)$?
Thanking you!
It seems that the corresponding PDE for marginal probability distribution does not exist because the processes x(t) and y(t) are not Markovian anymore.