Consider a $d$-dimensional stochastic process $X_t$ having density $\rho(X_t,t)$ at time $t$. Suppose the drift term $b$ of the process depends on the density $\rho = \rho(X_t,t)$: $$ dX_t = b(X_t,\rho) dt + dW_t$$
(this is called McKean-Vlasov equation, from mean field theory).
The corresponding Fokker-Planck equation (FPE) describing the evolution of the density is $$ \partial_t \rho = \nabla \cdot (b(X_t,\rho) \rho) + \Delta \rho.$$
If the $i$th component of the drift is $b_i(X_t,\rho)$ then we can also write out the $i$th component of the stochastic process and the FPE: $$\partial_t \rho_i = \nabla \cdot (b_i(X_t, \rho) \rho_i) + \Delta_\rho.$$
I asked this question a while back: given $d$ 1-dimensional FPEs with drift $b_i, i=1,\dots,d$, can we couple those into a $d$-dimensional FPE? I answered it and I think the answer is yes: but now I wonder if the stationary distribution for each of the $i$th 1-dimensional FPEs can be combined to become the stationary distribution of the $d$-dimensional.
This question is a bit different from the previous one, since the drifts are coupled here. If the drifts were not coupled I think the result would follow.