I'm working through First Steps in Random Walks by Klafter and Sokolov and I'm stuck at Exercise 3.5.
The Question: Exercise 3.5. Show that the PDF of the particle's displacement in the ultra slow CTRW (Exercise 3.4) follows a two sided exponential pattern $$P(x,t)=\frac{1}{2W(t)} \exp \left(-\frac{|x|}{W(t)}\right)$$ Find the temporal dependence of the width (W(t)) of this distribution.
Hint: Follow the procedure leading to Eq. 3.12 (in the textbook) for the normal case and perform first the inverse Laplace transform by noting that the corresponding $P(k,s)$ has the form $\frac{1}{s}L\left(\frac{1}{s}\right)$ with $L(x)$ being a slowly varying function (depending on $k$).
The question expects the use of Tauberian theorems: $$f(t)\cong t^{\rho-1} L(t)$$ $$f(s) \cong \Gamma(\rho)s^{-\rho} L(1/s)$$
My sticking point:I've got as far as getting an expression for $P(k,s)$:
$$P(k,s)= \frac{1} {s(1+\frac{\sigma^2 k^2}{2}(\beta -1) ln^{(\beta-1)}(\frac{1}{s}))}$$
I'm stuck at the Laplace transform for $P(k,s)$. All attempts I've made results in complete nonsense. Can anyone suggest how to proceed? Thanks.