# I have this linear PDE...

Hi,

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.

Thanks

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 $Y = \nabla^{\perp} V$.