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

Article type: Research Article

Authors: Kang, Junjun | Tang, Yanbin^{; *}

Affiliations: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China

Correspondence: [*] Corresponding author. E-mail: tangyb@hust.edu.cn.

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 ut−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 ut−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

DOI: 10.3233/ASY-171410

Journal: Asymptotic Analysis, vol. 102, no. 1-2, pp. 55-70, 2017