Since you have a handle on the Process instance, you should be able to use process.destroy() when you stop your service. The Logcat processes are running independently of your app, so they will continue running after your app's Service stops running unless you terminate them.
If you just want to clean up the processes on your device while testing the app, you can run adb shell to get a shell prompt on the device, and then use kill <PID> to terminate a process.
The Process documentation also specifies that you should use ProcessBuilder to set up and run the process:
ProcessBuilder pb = new ProcessBuilder("logcat", "-v", "time", "-s", arg);
Process process = pb.start();