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## MOMENT CALCULATION FOR TIME-FRACTIONAL DIFFUSIONS

The Fourier-Laplace transform of $D^\alpha p(x,t) = D_{\alpha,2} \Delta p(x,t)$ has the form

$p(q,s) = \frac{1}{s+ D_{\alpha,2} s^{1-\alpha} q^2}$

So the moments can be calculated by $E[X^k] = (-i)^k \phi_X(0)$ by taking successive derivatives and letting $q\rightarrow 0$.  It is easy to check this way that these have mean zero.

Just use Wolfram Alpha to compute $d^4/dq^4 (1/(s+D s^{1-a}q^2))$ and find that the only term without a factor of $q$ in the numerator is $24 D^2 s^{2-2 a}/(Dq^2 s^{1-a} + s)^3$.  Let $q\rightarrow 0$ and take an inverse Laplace transform to find the fourth moment.

$E(X^4) = L^{-1} ( 24 D^2 s^{-2a-1})$

Then one uses the Laplace transform pair $x^\nu \rightarrow \Gamma(\nu+1) p^{-\nu-1}$ to determine the constants.