2 random fields $b$ and $c$ are derived from random field $a$ by
$b=\nabla^2a\equiv(\partial_{xx}+\partial_{yy})a $
and
$c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$.
(all fields are 2-dimensional, $a$ and $c$ are real while $b$ is complex)
The 2-point correlation function of $b$: $\langle bb\rangle(x)$ is known. What is the 2-point correlation function of $c$?
What about higher order correlation functions?
This problem originates from astronomy. There exists a result at 2-point level. But the derivation process involves either integral of product of two Bessel functions or surface integral on a complex plane with poles. We cannot understand the mathematics involved rigorously, either can we generalize the result to higher order statistics.
Is there any established mathematical theory related to this? Thanks a lot!
