Hi,
I cannot find ApplicationClass in my help

But I'm pretty sure you can retreive his windows handle. Later when you wants to check if this particular instance of Mappoint is still running you can probably do it this way:
Code:
private bool IsThisOneRunning(IntPtr h)
{
Process[] processes = Process.GetProcessesByName("MapPoint");
foreach (Process proc in processes)
if (proc.Handle == h)
return true;
return false;
}