The Fokker-Planck equation for several variables is :
$\frac{\partial W}{\partial t} = L_{FP}$L_{FP}W$
where
$L_{FP} = -\frac{\partial}{\partial x_i}D_i({x})+\frac{\partial^2}{\partial x_i \partial x_j}D_{ij}({x}).$
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 $W_{st}$
$L_{FP} W_{st} = 0$,
may exist.
If one solves the above equation, a possible stationary solution can be
$W_{st} =\frac{a}{D_{ij}}exp(\int^{x_j}_0 \frac{D_i}{D_{ij}} dt_j)$
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
$W_{st} ={\frac{a}{D_{11}}exp(\int^{x_1}_0 \frac{D_1}{D_{11}} dt_1)+\frac{a}{D_{12}}exp(\int^{x_2}_0 \frac{D_1}{D_{12}} dt_2)+\frac{a}{D_{21}}exp(\int^{x_1}_0 \frac{D_2}{D_{21}} dt_1)+\frac{a}{D_{22}}exp(\int^{x_2}_0 \frac{D_2}{D_{22}} dt_2)}$.
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?
