The Fokker-Planck equation for several variables is :
$$\frac{\partial W}{\partial t} = L_{FP}W\qquad(1)$$
where
$$L_{FP} = -\frac{\partial}{\partial x_i}D_i(\{x\})+\frac{\partial^2}{\partial x_i \partial x_j}D_{ij}(\{x\}).\qquad(2)$$
The summation convention for Latin indices is used here. The drift vector $D_i$ and the diffusion tensor $D_{ij}$ generally depend on the N variables $x_1,...,x_N = \{x\}$. The Fokker-Planck equation is an equation for the distribution function $W(\{x\},t)$.
According to [Risken 1989 ch6], If drift & diffusion coefficients do not depend on time & $D_{ij}$ is positive definite everywhere & if the drift coefficient has no singularities, a stationary solution $W_{st}$
$$L_{FP} W_{st} = 0,\qquad(3)$$
may exist.
If one solves the above equation, a possible stationary solution can be
$$W_{st} =\frac{a}{D_{ij}}\exp\left(\int^{x_j}_0 \frac{D_i}{D_{ij}}\mathrm dt_j\right)\qquad (4)$$
where a is a normalization constant. Now I want to expand this probability distribution for $i=1,2$. If I use the Einstein summation convention, it becomes
$$\begin{split}W_{st} =\left\{\frac{a}{D_{11}}\exp\left(\int^{x_1}_0 \frac{D_1}{D_{11}}\mathrm dt_1\right)+\frac{a}{D_{12}}\exp\left(\int^{x_2}_0 \frac{D_1}{D_{12}}\mathrm dt_2\right)+\right.\\\left.\frac{a}{D_{21}}\exp\left(\int^{x_1}_0 \frac{D_2}{D_{21}}\mathrm dt_1\right)+\frac{a}{D_{22}}\exp\left(\int^{x_2}_0 \frac{D_2}{D_{22}}\mathrm dt_2\right)\right\}\end{split}\qquad (5)$$
It seems very strange to me. Is it a really correct probability distribution or I made a mistake somewhere? And if it is correct how can I normalize it? Can anyone help?
(I will use the Einstein summation convention and $\partial_i = \frac{\partial}{\partial{x_i}}$)
The stationary solutions fulfill $L_{FP} W_{st}=0$ as explained. This can be rewritten as $\partial_i J_i =0$ where we have introduced the current $$J_i = D_i W_{st} - \partial_j D_{ij} W_{st}.$$ Many FP equations allow for a stationary solution where the current vanishes $J_i=0$ (and not only the divergence is equal to 0). Assuming that, we obtain $D_i W_{st} = \partial_j D_{ij} W_{st} = (\partial_j D_{ij}) W_{st} + D_{ij} \partial_j W_{st} $ which is equivalent to $$ \partial_j \log W_{st} = (D^{-1})_{ji} (D_i - \partial_k D_{ik}).$$ Integrating this equation, we obtain $$ W_{st}(x) = c \exp\left[\int^x dx'_j (D^{-1})_{ji} (D_i -\partial_k D_{ik}) \right]$$ with $c$ some normalization constant. So the last two equations of the questions are in fact wrong.