While playing the game, possibly right as you started, you may have noticed a lot of persistent stutter. This comes from the framerate not stabilising at any number, which results in skipped frames. The annoying part is, changing the video settings does not affect this issue. Looking at the Task Manager you might even see that the game doesn't use all the resources your computer has to offer (For example, my CPU usage never exceeded 30%, Memory stayed at 40% and GPU didnt even reach 20% despite the 3D Animations). It appears to be an optimization issue from when the game was ported to steam.

Assuming you are experiencing the same issue as I was (that is, not worse ones like constant crashes etc.), a simple solution I found was to set the priority of the game higher in the Task Manager settings.
To do this, you:

1. Run the Game
   
2. Open the Task Manager, either via the Windows Start-Menu or by pressing CTRL+ALT+DEL and choosing Task Manager.
   
3. Switch to the Details Tab and select Heavy Rain.exe
   
4. Right click and select Priority
   
5. Assuming you never changed these settings before, it should be set to Normal by default, like most other applications. Set it to a higher value (I personally used High)

There is no need to restart the game, just close the Task Manager and tab back into the game to test the framerate. Chances are you will notice a not so insignificant improvement, that is, less stutter.

If doing this helped your performance, that's great! Now you can enjoy this somewhat older game.
If the stutter didn't go away entirely or at all, then I'm sorry I couldn't help out more.

I'm no expert, but I do have a theory as to why this (specific) issue occurs:
The computer system normally runs several, if not many, applications at once, a lot with the same priority (You can check for yourself in the Task Manager). When you're actively interacting with an application, say a game, the priority of this application is higher than the ones in the background, so the system focuses resources on the active window. I'm thinking Heavy Rain does not properly utilise this, so the computer often switches focus during gameplay, resulting in lost frames. By setting the priority higher, this no longer occurs.

But who knows, I might be wrong. I'd be happy to hear a professionals opinion on this.
Thank you for reading this humble guide and I hope I could help you to enjoy this game.