Estimates on Green functions and Poisson kernels for symmetric stable processes

Abstract

One of the most basic and most important subfamily of Levy processes is symmetric stable processes. A symmetric α-stable process X on Rn is a Levy process whose transition density p(t , x − y) relative to the Lebesgue measure is uniquely determined by its Fourier transform ∫ Rn e ix ·ξp(t , x )dx = e−t|ξ| α . Here α must be in the interval (0, 2]. When α = 2, we get a Brownian motion running with a time clock twice as fast as the standard one. Brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics. Thus it has been intensively studied. In this paper, symmetric stable processes are referred to the case when 0 < α < 2, unless otherwise specified. In the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes. Stable processes are now widely used in physics, operations research, queuing theory, mathematical finance and risk estimation. In some physics literatures, symmetric α-stable processes are called Levy flights, and they have been applied to a wide range of very complex physics issues, such as turbulent diffusion, vortex dynamics, anomalous diffusion in rotating flows, and molecular spectral fluctuations. In mathematical finance, stable processes can be used to model stock returns in incomplete market. For these and more applications of stable processes, please see the interesting book [14] by Janicki and Weron and the references therein and the recent article [15] by Klafter, Shlesinger and Zuomofen. In order to make precise predictions about natural phenomena and to better cope with these