Potential theory of generalized hyperbolic processes.

摘要

Let Yt be a rotationally invariant generalized hyperbolic process in Rd , d ≥ 3. Yt can be obtained by subordinating Brownian motion with a generalized inverse Gaussian subordinator Tt. We introduce generalized inverse Gaussian and generalized hyperbolic processes in chapter 1. In chapter 2, we study the asymptotic behaviors of the Green function of Y t near zero and infinite, and jumping function of Yt near zero. We prove that Harnack inequality is valid for nonnegative harmonic functions of Yt. In chapter 3, we show that the rotationally invariant generalized hyperbolic processes in a bounded C1,1 open set D can be obtained from rotationally invariant Cauchy processes in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable, and the sharp estimate of the Green function of Yt in D is given. In the last chapter, we show boundary Harnack principle holds true for rotationally invariant generalized hyperbolic processes.