Fourier Transform of Additive White Gaussian Noise?
You calculate the Discrete Fourier Transform of Additive White Gaussian Noise like this.

1) Fill a time vector with samples of AWGN
2) Take the DFT

The result will appear to be random.

How you interpret the resulting samples is another matter. Why are you taking the FT of AWGN in the first place?

It may be that you want to know how much energy is in each bandwidth of the AWGN, assuming it's being generated by an ergodic process. This will require some more processing. You will want to break your time record up into overlapping chunks, window each to avoid spectral leakage, then FT each separately. You will want to calculate the power in each bin, and then sum those powers over each bandwidth you want to examine, then average the power per bandwidth per frequency over all the transforms to reduce the variance.

Now you will start to get a result that describes the long term power spectrum of your AWGN.

If you want something else out of your FT processing of AGWN, then you will need to do something different with the results.

If you really mean FT, rather than DFT, you would be better off taking the question to the mathematics site.