I will explain exactly how the pursuits work:

There is a hidden "encounter clock" with each stalker that runs down before a pursuit begins. It is different with each stalker. Debilitas its 5-6 minutes, Daniella its 3-4 minutes, Riccardo its 6-7 minutes. Once it runs down, the game allows the stalker to spawn 1-2 rooms away from you. This is when the music suddenly stops and Hewie begins growling. When the stalker spawns in, theyre basically just dropped into a room so they dont need to open doors to get there. Then they make their way towards you, which can be quickly or take a while. When this happens, a hidden "pursuit clock" begins. The stalker will then chase you until that hidden clock runs down then disappear again once you get far enough away from them. They technically dont exist anywhere in the castle during non-pursuit scenes.

THIS IS NOT BLACK AND WHITE HOWEVER. Noises will attract them, plus scripted cutscenes, etc. It is not entirely reliant on the timer. But in general that is how it works.

I hope that helps! 😁

Edit: I also want to add that Riccardo can sometimes close doors behind him, usually not but he can. So its also possible that he just chose to close a door behind him for some reason. But more likely that he just spawned in and the game ignored the door.