## Asking for a Fourier inverse transform, which is related to stable laws

Dear friends,

Denote the function

$$G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;.$$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of symmetric stable laws. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints or references.

Anand

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I think you may want to look at this MSE question. Amusingly enough, this question was to prepare a lecture for when the asker was covering my class last fall. This kind of argument is mentioned in Durrett's Probability: Theory and Examples in Chapter 3.

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 Thanks Professor BSteinhurst, I will have a look. As far as I know, the best reference for stable laws is Zolotarev's book (one-dimensional stable distributions). In that book, $a\in ]0,2]$. – Anand Mar 16 2012 at 10:07

Precise estimates of the function $G_\alpha$ (also in the multi-dimensional case), its derivatives, and more complicated functions appearing as fundamental solutions of parabolic pseudo-differential equations with homogeneous symbols, can be found in Chapter 4 of the book:

S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser, Basel, 2004.

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 Thanks Professor Kochubei, I have your book at my hand since last time you answered my question giving me the same reference. I will have a look. Thank you very much! :-) – Anand Mar 16 2012 at 9:54 Dear Professor Kochubei, I will take some time to learn your book. I am wondering whether it's possible to briefly answer the question how $G_a(x)$ decays at infinite? Thanks a lot! – Anand Mar 16 2012 at 10:10 $|G_\alpha (x)|\le C(1+|x|)^{-n-\alpha}$ where $n$ is the space dimension. – Anatoly Kochubei Mar 16 2012 at 16:07 Dear Professor Kochubei, thanks a lot for your help! :-) – Anand Apr 3 2012 at 20:36

See the article "Some theorems on stable processes" in Trans. Amer. Math. Soc. vol. 95 (1960), pp. 263–273, by Blumenthal and Getoor. Apparently the computation of this Fourier transform is due to Polya in "On the zeros of an integral function represented by Fourier's integral", Messenger of Math. vol. 52 (1923), pp. 185-188.

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 Thanks Professor Abdelmalek Abdesselam, I think that these papers deal with $\alpha\in ]0,2]$. Thanks. :-) – Anand Apr 3 2012 at 20:39 @Anand: the paper by Blumenthal and Getoor has the answer to your question for any positive alpha not just for those in the interval ]0,2]. Look carefully at the statement of Theorem 2.1 – Abdelmalek Abdesselam Apr 22 2012 at 19:25