Summary:

To find the PID of a running UNIX/XWindow process:

$ xprop _NET_WM_PID

If that fails:

$ ps -ww -fp <PID>
$ pwdx <PID>

Detailed explanation:

So How can you find the PID (not Window ID!) of a running UNIX/XWindow process?

Method 1 (simple):

$ xprop […]