Took the controller apart, to see if there was something wrong inside the controller. Cleared away any dust with canned air, and examined it thoroughly. There wasn't anything necessarily wrong with the hardware itself that I could see, and looked relatively clean, even before I dusted it. Put it all together again, booted up the PS4 and went straight to the web browser to see the results.
Obviously, it didn't fix it, seeing as the mouse cursor still drifted upwards. The problem is the software inside the controller itself. And since it's clearly obvious that support doesn't want to answer or acknowledge me, I'm just going to assume that this problem cannot be fixed.
Support, and Sony, please include a way to recalibrate the PS4 controller within the console itself. I would believe that would solve this immediately.
There is absolutely no solution to this problem. I've open my controller about 7 times, replacing the analog itself and a bunch of other tomfoolery (didn't work, unless the replacement was as messed up as the last one); the problem is most likely the sisterboard inside the controller itself. You can try opening the controller yourself and taking some canned air to it if it's just dirty from dust or something, maybe replacing the analog if you're skilled in soldering and have another one available.
Otherwise, your best bet is return the one you have now, and get another controller. There is no solution for this problem beyond that.
I am having this issue with the left analog stick on the Dual Shock 4 controller I received at console launch in the US. It's very bad and I won't be able to play any more games until I get a new controller. I've tried resetting the controller, I've tried using it only while plugged into the console with the usb cable.
The left stick just seems to alternate beteen jittery response or intermittent non-responsiveness. Noticed this while playing through Infamous, but it's occuring on all games. You really canot notice this within the menu system or on the dashboard, this occurs mostly within games.