The screen woke up at 4am or 5am at night, and had been on ever since.
How can I find out the process that was responsible for screen-wake ups?
The device is Galaxy S III running Android Jellybean 4.1.2.
Download the GSam Battery Monitor (formally knows as Bad Ass Battery Monitor) it will keep track of the number of times and for how long each app keep the device awake.
Open GSAM and go to the App Sucker view (icon on the lower left). Then change the view to Time Held Awake. To see the number of time an app held the device awake press on the app.
Check "Wakelock Detector" - WLD at GooglePlay (disclosure: I am the author)
Choose "Full wakelocks" option to see which apps are using screen wakelocks!
Currently running processes are shown in Green color, which makes it easy to recognize the suspicious app.
Here is a screenshot, where the Ad Notification was detected. It was waking up my screen randomly.
In Order to determine, which process causing the WakeLock, you can use betterBatteryStats from XDA.
Its proven over periods, the App provides clear list of what are all process causing the wake lock and their count as well.
Hope it helps!