Use WMI to get process id by name and user. The problem is that Win32_Process.GetOwner requires SeDebugPrivelege, and .Net framework does not let you adjust priveleges.

I'm using COM interop to get there...

private SWbemObject GetProcess(string image, string owner)
		{
			WbemScripting.SWbemLocator locator = new WbemScripting.SWbemLocator();
			WbemScripting.SWbemServices service = locator.ConnectServer("", @"rootcimv2", "", "", "", "", 0, null);
			service.Security_.Privileges.Add(WbemScripting.WbemPrivilegeEnum.wbemPrivilegeDebug, true);

			string strQuery = string.Format("SELECT * FROM Win32_Process WHERE name = '{0}'", image);

			WbemScripting.SWbemObjectSet searcher = service.ExecQuery(strQuery, "WQL", (int)WbemScripting.WbemFlagEnum.wbemFlagReturnImmediately, null);
            
				foreach (SWbemObject proc in searcher) 
				{
					string s = proc.Properties_.Item("ProcessID", 0).get_Value().ToString();

					// don't check owner, just return this process!
					if(owner == "")
						return proc;

					SWbemObject result = null;
					string proc_path = proc.Path_.Path;

					try 
					{
						result = service.ExecMethod(proc_path, "GetOwner", null, 0, null);
					}
					catch(Exception ex)
					{
						continue;
					}
					object retval = result.Properties_.Item("ReturnValue", 0);
					if(result.Properties_.Item("User", 0) != null)
					{
						string user = result.Properties_.Item("User", 0).get_Value().ToString();
						string domain = result.Properties_.Item("Domain", 0).get_Value().ToString();

						if(user == owner)
							return proc;
					}
					else
					{
						MessageBox.Show(this, string.Format("proc.InvokeMethod returned {0}", retval.ToString()));
					}
				}
			return null;
		}