Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes

Sihun Jo, Minsuk Yang

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional integral to the contour integral representation. We then express the integrand as a combination of gamma functions so that we can easily find all poles of the integrand. We obtain various asymtotics of the kernels by using Cauchys Residue Theorem with shifting contour integration. As a byproduct, exact coefficients are also obtained. We apply this method to general Lévy processes whose characteristic functions are radial and satisfy some regularity and size conditions. Our approach is based on the Fourier analytic point of view.

Original languageEnglish
Pages (from-to)203-230
Number of pages28
JournalPotential Analysis
Volume40
Issue number3
DOIs
Publication statusPublished - 2014 Jan 1

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Precise Asymptotics
Asymptotic Approximation
Integrand
kernel
Residue Theorem
Symmetric Stable Processes
Contour integral
Mellin Transform
Gamma function
Complex Analysis
Fractional Derivative
Characteristic Function
Integral Representation
Pole
Express
Regularity
Singularity
Decay
Transform
Coefficient

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

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Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes. / Jo, Sihun; Yang, Minsuk.

In: Potential Analysis, Vol. 40, No. 3, 01.01.2014, p. 203-230.

Research output: Contribution to journalArticle

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