I’ve only just started playing with appscript properly. It is übercool, but I clearly don’t grok everything it can do just yet!

isrunning = app('System Events').processes['AppName'].exists()

Note that with the above approach (which uses the by-name reference form), process names are case-sensitive for some reason. If you want case-insensitivity, using the by-test reference form seems to work:

isrunning = app('System Events').processes[its.name == 'AppName'].exists()

Alternatively, you can use the aem module to check for a running process based on the application’s path:

from aem import Application
isrunning = Application.isrunning('/path/to/some.app')

aem also includes a findapp module that you can use to locate an application by name, creator type or bundle id. The appscript documentation has more info on this.