Derive the 2-D analogue of the Laplace Dispersal Kernel using RDE

Question

I found an interesting problem. I'm looking at the Laplace Dispersal Kernel for 1 dimensional dispersal behavior. And I wonder what happens in two dimensional world? I managed to find the limiting RDE but stack on deriving the actual kernel. To simplify things, I do it for all parameters constant in space and time. Still having troubles deriving the result. Would appreciate any advice!!!

Answer 1

Not sure if anyone is interested but I found the solution. For a symmetric kernel, I can set q=0 and integrate both parts. After that we get the modified Helmholtz equation for 2-dimentional case. The Green's function for that equation is my kernel, so I get a solution in terms of Hankel function which can be replaced by modified Bessell function. That's it. :) This was a fun one.