The PS4 is confirmed to have a separate, dedicated chip for AV compression/decompression. For Sony, this is likely to be an off-the-shelf chip, (or based on an off-the-shelf chip) from the Digital Imaging division (which makes world-class video cameras, ect.) of their own company. So they probably have a lot of designs to choose from for the best fit in the PS4.
These dedicated chips are purpose-built, highly-efficient streaming processors that have their own fast local store on the video chip itself (SRAM) or next door (DRAM) for the reference frame storage, ect. To give you a frame of reference of how efficiently and fast these processors chew through the data, a typical chip of this type for 1080p/60 H.264 processing can get away with having around 256kb of this type of local store. (In fact, if you read any number of white papers on various processors of this type, you can see this is actually a larger footprint than many actually use. Remember, these chips don't need to work on a whole frame at a time. They operate super-fast, so the task is broken up into the tiniest pieces.)
From there, it's a simple write to the HDD to store the file for later.
Think about it this way...in an HD camera like the one in your nicer smartphones, a GoPro or Sony's SONY AS15 "action camera," do you think they have a whole bunch of random memory on board? Of course not.
Knowing how this works, and knowing that they have this dedicated chip aboard, it's reasonable to assume that any OS activity reserved for this function would be extremely negligible to the OS/memory footprint.