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The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$

where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is the stationary bit of a Fokker-Planck equation derived from a generally complex Langevin equation, i.e. $x = y+iz$.

I was wondering if anyone knew of a standard table/textbook that details the solution or how to solve PDEs like the above? With no complex part ($z=0$) the solution is a Gaussian (with suitable restrictions on the coefficients).

It is possible to let $A$ or $B$ be zero and the ensuing equation is still meaningful.


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up vote 1 down vote accepted

If you are looking for solutions in the whole plane, take the Fourier transform. You get a first order PDE which you can then analyze by the method of characteristics.

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This is just a comment on Rafa's answer, but I don't seem to find the way to make comments. The PDE is not necessarily elliptic (take e.g. A = 1 and B = -1, resulting in the wave equation). If A and B are (identically) zero then the equation just says that the divergence of the vector field $Y(y, z)$ with entries $Y_1(y,z) = (Cy - Dz)P(y, z)$ and $Y_2(y, z) = (Dy + Cz)P(y, z)$ is zero. If the domain is simply connected, then that is equivalent to saying that there exists a scalar function (a potential) $V$ such that $Y = \nabla^{\perp} V$.

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Ooo, that's right, I agree – guacho May 30 '13 at 8:36

This is an elliptic operator. I think that Chapter 6 of Evans's book "Partial differential equations" is appropriate if the domain is bounded. In the whole plane you can use Fourier techniques, isn't it?

I hope this help you

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