D-dimensional Lévy flights: Exact and asymptotic

T. M. Garoni, N. E. Frankel

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    5 Citations (Scopus)


    The analytic and asymptotic properties of the spherically symmetric d-dimensional Lévy stable probability density function, pα d(r), are discussed in detail. These isotropic stable probability density functions (pdfs) are analogous to the one-dimensional symmetric Lévy stable pdfs previously studied by the present authors [J. Math. Phys. 43, 2670 (2002)]. We construct a hypergeometric representation of pα d(r) when α is rational, and find a number of new representations of pα d(r) in terms of special functions for various values of d and α. A recursion relation is found between pα d(r) and pα d+2(r), which, in particular, implies there exists a simple map between pα 1(r) and pα 3(r). As in our previous paper, we discuss the properties of pα d(r) for both the cases α≤2 and α>2. We demonstrate the existence of intricate exponentially small series in the large r asymptotics of pα d(r) when α is an integer, which are dominant when α is even. We explicitly construct this beyond all orders expansion of pα d(r) for arbitrary integral α and d.

    Original languageEnglish
    Pages (from-to)5090-5107
    Number of pages18
    JournalJournal of Mathematical Physics
    Issue number10
    Publication statusPublished - 1 Oct 2002

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