[Math] Fourier transform of a random variable

fourier analysisintegral-transformsrandom variables

During my research i'm dealing with a stochastic partial differential equation.
The random term appearing in my equation is a tensorial random variable:
Which is a gaussian random variable completely defined by the first two moments:
$<\boldsymbol{\sigma}(\boldsymbol{x},t)\boldsymbol{\sigma}(\boldsymbol{x^{'}},t^{'})>=\frac{2k_{B}T}{\mu}(\delta_{i,j}\delta_{k,l}+\delta_{i,l}\delta_{j,k}) \delta(\boldsymbol{x}-\boldsymbol{x}^{'}) \delta(t-t^{'})$

$i,j,k,l$ denote the components of the tensor.

The tensorial random variable is essentially a white noise in space and time.
I'm reading a paper in which the authors take the spatial fourier transform of this random variable.
They report that the fourier transformed second moment is:


In this formula $\boldsymbol{k}$ is the wave vector.

I don't really understand how this result is obtained, so if anyone could help me with the derivation it would be great, thank sin advance.
I hope the question is clear, if you think it is not, please let me know i'll add more details.

Best Answer

The Fourier transform of a delta function is a constant. $$FT[\delta(\mathbf y - \mathbf z)] = e^{-2\pi i \mathbf k \cdot \mathbf z}$$ $$\Rightarrow FT[\delta(\mathbf y)] = e^{-2\pi i 0} = 1$$ Now for your variance tensor let $\mathbf x^\prime = \mathbf x + \mathbf y$ which gives you $$ \langle \sigma(\mathbf x, t)\sigma(\mathbf x + \mathbf y, t^\prime) \rangle = \frac{2k_bT}{\mu}(\delta_{i,j}\delta_{k,l} + \delta_{i,l}\delta_{j,k} )\delta(\mathbf y)\delta(t-t^\prime)$$ Thus Fourier transforming this wrt space ($\mathbf y$) gives back $$ \frac{2k_bT}{\mu}(\delta_{i,j}\delta_{k,l} + \delta_{i,l}\delta_{j,k} )\delta(t-t^\prime)$$

As a general comment (unless I've misunderstood something) the notation seems somewhat untidy, either there should be a sum over indexes on the RHS or there should be corresponding indexes $i,j,k,l$ on the LHS, since this doesn't appear to be using Einstein's summation convention.

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