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Abstract

In this paper we consider the asymptotical behavior of the density function of one dimensional nonsymmetric layered stable processes via Lévy–Khinchin exponent. For the corresponding parabolic partial integro-differential equation $u_{t} - L [u] = 0$ generated by the Lévy operator L, the infinitesimal generator of the layered stable process, we first develop a new analytical method to instead of the usual probability method, then we give the long-time and the short-time asymptotical behavior of the solutions to the corresponding Cauchy problem of the equation $u_{t} - L [u] = 0$ respectively, and these imply the asymptotical behavior of transition density. Finally we localize the parabolic partial integro-differential equation to the bounded domains and give the error estimates due to the localization.

Keywords

Transition density Lévy–Khinchin exponent layered stable process asymptotical behavior Gårding’s inequality

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Asymptotical behavior of partial integral-differential equation on nonsymmetric layered stable processes

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