Boot problems may happen with any device even if you don’t try harm the device. Like Galaxy S2 had boot problems
right after the Firmware updates. Or it may happen when you try to play
with the device’s software/firmware and it gets bricked. And more such
possibilities while/after installing Custom ROMS.
Fortunately, This boot problem was not with my SGS3 but with a
XDA-user who was searching for solution. He tried to install SiyahKernal
and his Galaxy S3 got stuck at welcome screen. The same case happened
with S2 while installing the official updates on stock ROM(please
note).
The process to recover your S3 is same as we did for S2 but the files
required are for it are different. You might think of saving the data
on your device so let’s first start with Soft Reset(read
below) which cleans the device and the data remains intact. If the
device isn’t booting after the soft reset then you are left with two
options – Hard Reset and Re-install the Stock ROM. In
most of the possibilities you will have to re-install the stock ROM
which is not a lengthy and complicated process which many users are
afraid of.
How to Soft Reset Samsung Galaxy S3?

• Take out the battery.
  
• Reinsert the battery.
  
• Press and hold the power button for 10 seconds.
  

How to Hard Reset Samsung Galaxy S3?

• Switch off the device.
  
• Press and hold Volume Up + Home Button + Power Key.
  
• After 2-3 seconds, release the Power Key.
  
• You will enter boot screen, Select Wipe Data / Factory Reset.
  
• Select Yes.
  

How to Re-install Stock ROM on Samsung Galaxy S3?

Forget the crappy Samsung Kies, the PC suite for Samsung, It cannot
solve the issue. And for me, It’s junk, it’s useless. What you need is
Odin, .PIT File and the Stock ROM. Be careful to choose the Stock ROM
for your S3 as it varies from country to country and carrier to carrier.
Download these files:

• Odin
  
• .PIT File (GT-I9300_mx_20120329.PIT)
  

Process starts here:

1. Pull out the battery and reinsert.
   
2. Press and Hold Volume Down + Home + Power Button until you see a warning!!
   
3. Press Volume Up button.
   
4. Your device will now enter the download mode.
   
5. Connect your phone to the PC.
   
6. Let it install some drivers.
   
7. Open Odin and click on PIT
   
8. Select that .PIT file you downloaded.
   
9. Check the box – Re-partition and Click Start.
   
10. After the successful attempt to re-partition the device, Flash with new firmware(which you downloaded from SamMobile)
    
11. Browse the downloaded firmware to the PDA section in Odin.
    
12. Don’t check the re-partition box this time.
    
13. Click Start and wait.
    
14. As soon as the process completes, your phone will boot. and it won’t stuck at welcome screen.