I am looking for the general stationary solution of the Fokker-Planck equation in two or more dimensions. I know that for the one-dimensional case, given a drift $F(x)$ and position-dependent diffusion $D(x)$, the FP equation is $$ \frac{\partial p(x,t)}{\partial t} = - \frac{\partial}{\partial x} [F(x)p(x,t)] + \frac{\partial^2 }{\partial x^2}[D(x) p(x,t)] $$ and the stationary solution is $$ \pi(x) = \frac{N}{D(x)} \exp\left(\int^x_{-\infty} \frac{F(x)}{D(x)} \mathrm{d}x \right) , $$ with $N$ normalization constant.
Question
Is it possible to generalize this result for the $N$-dimensional case or at least the 2-dimensional case with a position dependent diffusion matrix D(x,y)? $$ \frac{\partial p(x,y;t)}{\partial t} = - \frac{\partial}{\partial x} [F_x(x,y)p(x,y;t)] - \frac{\partial}{\partial y} [F_y(x,y)p(x,y;t)] + \frac{\partial^2 }{\partial x^2}[D_{11}(x,y) p(x,y;t)] + \frac{\partial^2 }{\partial y \partial x}[D_{12}(x,y) p(x,y;t)] + \frac{\partial^2 }{\partial x \partial y}[D_{21}(x,y) p(x,y;t)] + \frac{\partial^2 }{\partial y^2}[D_{22}(x,y) p(x,y;t)] $$
If this is not possible, does it exist at least the stationary solution for the case with non-position dependent diffusion matrix, i.e. D(x,y)=D, with $D_{11}\neq D_{12}\neq D_{21}\neq D_{22}$.?