Hello, I have the same identical notebook and I had the same issue...
Unfortunately it's a common issue that can happen to those who have a nVidia 8-series graphics chip, especially 8400 and 8600.
The only safe and official way to fix the issue would be to send an RMA to Sony in order to have the motherboard changed, since the nVidia 8400M chip is soldered to the motherboard.
However I managed to solve the problem unofficially, with heat.
More specifically it can happen that, whenever artifacts are shown on screen due a faulty graphics card, this card can be fixed by putting it into an oven for about 10-15 minutes at about 180°C or locally heating the chip to melt again the soldering joints.
It's a trick which doesn't make damage to the card and worked for me a couple of times, but I can't guarantee it to work forever...
A week ago, I took my VGN-AR51E and unscrewed the plastic panel under it (it came off very easily after taking all the screws), then I unscrewed the aluminum heatspreader placed on the motherboard and, with a hairdryer at maximum heat and power, I heated for 15 minutes the bare chip above the ram slots, then other 15 minutes on the chip up on the right (which is the graphics chip).
After that, my notebook started working perfectly again and I've been also playing videogames since.
This is not a joke and there are also tutorials on the net about this trick.