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?