In the Wikipedia article on Brownian noise, the Fourier transform of Brownian noise is determined. How is that Fourier transform defined? It seems it is a non-random quantity there, so it is not simply a pathwise Fourier transform. Since in the same article, it is said that white noise has constant Fourier transform and I know that the dirac delta has a constant Fourier transform, I actually guess that "Fourier transform of a (stationary) random process" means "Fourier transform of the covariance function of the random process". Is that true? Is there intuition behind this definition? Also, if this is the definition, how does the reasoning that $\mathcal{F}(dW(t)/dt)(\omega)=i\omega \mathcal{F}(W(t))(\omega)$ still go through? References?

  • 2
    $\begingroup$ The linked article speaks of the power spectral density and power spectrum of a Brownian noise random process, not its Fourier transform. As you pointed out, it doesn't make sense to speak of a deterministic Fourier transform of a random process. $\endgroup$ – Jason R Oct 2 '15 at 14:20
  • $\begingroup$ Reiterating @JasonR's point: The Fourier transform of white noise is white noise. The power spectrum of white noise is a constant. $\endgroup$ – Peter K. Oct 2 '15 at 16:42
  • $\begingroup$ Also have a look at this answer to a related question. $\endgroup$ – Matt L. Oct 2 '15 at 19:16
  • $\begingroup$ reading up on the power spectral density answered my question. the article is simply missing an expected value, i guess $\endgroup$ – Bananach Oct 3 '15 at 8:00

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