I've been trying to find a solution to the following 2D, stationary Fokker-Planck equation:
$$\frac{\partial}{\partial x}\left[D_x \frac {\partial P}{\partial x} + P \frac{\partial U}{\partial x}\right] + \frac{\partial}{\partial y} \left[D_y \frac{\partial P}{\partial y} + P \frac{\partial U}{\partial y}\right]=0$$
in order to find the probability distribution $P(x,y)$ (I'm working with a brownian dynamics system at two temperatures and have been trying to work out the equilibrium distribution).
For the systems I'm working with, I can assume that
$$\frac{\partial U}{\partial x} = a_1 x + a_2 y $$ $$\frac{\partial U}{\partial y} = a_3 x + a_4 y$$
So far, I've tried tackling this with a separation of variables approach, e.g. assuming $P(x,y)=\Phi(x)\Psi(y)$
but it doesn't appear possible to separate the variables.
Is there another way standard way to approach this type of problem? If anyone could point me in the right direction as to anything else I might try, I'd be in your debt.
Thanks!
Edit:
There are no bounds on $x$ or $y$: $x,y \in (-\infty, \infty)$
The only condition is that:
$$ \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dy \; P(x,y) = 1$$
Here is a solution for the case $D_x = D_y = 1$.
In general, a solution of your problem is given by $P(x,y) = c \cdot e^{-U(x,y)/2}$. This is a multiple of a probability distribution if $U \to \infty$ fast enough as $x,y \to \infty$.
If the partial derivatives of $U$ are linear functions of the form given in your post, then necessarily $a_2 = a_3$. Now set $A = \begin{pmatrix} a_1 & a_2 \\ a_2 & a_4 \end{pmatrix}$ and write $\mathbf{x}$ for the column vector $(x,y)^T$. Your equation may now be written compactly in the form $$ \nabla \cdot \left( \nabla P + P \cdot A\mathbf{x} \right) = 0 $$ A solution is given by $$ P(\mathbf{x}) = c \cdot e^{-\frac{1}{2}\mathbf{x}^TA\mathbf{x}} $$ or written out in detail $$ P(x,y) = c \cdot e^{-(a_1x^2 + 2a_2xy + a_4 y^2)/2} $$ where $c$ is any constant. This has a finite integral if $A$ is positive definite, i.e. $a_1 > 0$ and $a_1a_4 - a_2^2 > 0$. In that case you can find a positive constant $c$ such that $P$ is a probability density.