Harnack Inequality For Subordinate Censored Stable Processes

Markov processes with discontinuous sample paths constitute an important fami-ly of stochastic processes in probability theory,and censored stable processes are the typical class of discontinuous Markov processes.Subordinating the censored stable process,it gets a new process which is called subordinate censored stable process.Har-nack inequality is extremely important in harmonic analysis and probability theory.The Harnack inequality for subordinate censored stable processes is studied in this paper.Let D be a bounded C1,1 open set in Rd with d ? 1.Let XD be a censored ?-stable process in D with ??(1,2).S is a ?/2-stable subordinator with Laplace exponent?,??(0,2).Subordinate the process XD by S,and note that YtD:=(XD)St.The Harnack inequality for YD is proved in this paper.Some preliminary knowledge of subordinate censored stable processes is intro-duced in the first chapter,including the conclusions of stable subordinator,the equiv-alent definitions of censored stable processes and the properties of harmonic function for censored stable processes.The relationship of potential operators of YD and XD is found out by using the above preliminary knowledge,then the excessive functions of YD and XD are in one to one correspondence and the harmonic functions of YD and XD are in one to one correspondence.The continuity of nonnegative harmonic func-tions for YD is given through the continuity of nonnegative harmonic functions for XD.At last,the Harnack inequality for YD is established by using the above conclusions.The last chapter shows the conclusions of this paper.The estimates of Green function and jump function of YD are also given in this chapter.